Polarization calculation optimization methods, systems, apparatus, and media for RCWA

Abstract

The application discloses a polarization calculation optimization method, system, device and medium for RCWA, and the method comprises the following steps: performing single-time environment initialization and operator cascade according to optical incidence parameters and optical geometric parameters of a multilayer grating structure to obtain a system control matrix; constructing an incident excitation vector under double polarization states respectively, and performing horizontal splicing on the incident excitation vector under the double polarization states to obtain a double-column joint excitation matrix; performing linear solution on two columns of right end items of the double-column joint excitation matrix in parallel to obtain a double-polarization forward field coefficient matrix; performing matrix-matrix multiplication in parallel based on the double-polarization forward field coefficient matrix to obtain a reflection coefficient matrix under the double polarization states; and assembling the reflection coefficient matrix under the double polarization states to obtain a Jones matrix. The application improves the overall efficiency of polarization calculation at the premise of ensuring the consistency of algorithm physical robustness and calculation accuracy, and also realizes balanced optimization of memory occupation and algorithm power consumption.

Description

Technical Field

[0001] This invention relates to the field of semiconductor optical measurement, and more specifically to a polarization calculation optimization method, system, apparatus, and medium for RCWA. Background Technology

[0002] Rigorous Coupled-Wave Analysis (RCWA) is a classic numerical method for solving the electromagnetic field distribution of periodic dielectric structures. Its core principle is to perform a Fourier harmonic expansion of Maxwell's equations in the spatial frequency domain, transforming the boundary value problem of the electromagnetic field in a non-uniform periodic medium into a matrix eigenvalue problem and a problem of solving a system of linear equations. For a multilayer periodic grating structure containing t layers of dielectric material, assuming the harmonic cutoff number is N, the total number of bases in the frequency domain is L = 2N + 1; since the electromagnetic field contains two mutually perpendicular transverse electric field components, the overall state space dimension of the system is 2L.

[0003] In engineering applications, to obtain the complete polarization response of the grating structure and solve the Jones matrix, it is necessary to calculate the complex reflection coefficient and transmission coefficient under p-polarized (TM polarization, polarization angle ψ=0) and s-polarized (TE polarization, polarization angle ψ=π / 2) incident radiation, respectively. Existing technologies generally employ a two-stage, independent serial call scheme: using p-polarized and s-polarized radiation as incident excitations respectively, two complete RCWA (Reference-Conducting Radiation-Warranty) calculations are executed. Each calculation sequentially completes the construction of the intrinsic layer Toeplitz matrix, eigenvalue decomposition, interlayer boundary recursion, system control matrix assembly, solution of linear equations, and backpropagation of field coefficients.

[0004] The aforementioned prior art has the following significant technical defects: First, the core computational steps are highly redundant. Core steps such as eigenvalue decomposition, interlayer boundary recursion, and system control matrix construction are determined solely by grating geometric parameters (period, layer thickness, boundary morphology) and material optical constants (refractive index, dielectric constant), and are completely independent of the incident polarization state; however, in two sequential calculations, the aforementioned complexity reaches O(L...). 3 The high-overhead steps are completely repeated, and the total computational cost of eigenvalue decomposition and matrix inversion under the M-layer structure is four times that of single-polarization computation, resulting in serious waste of computational resources.

[0005] Second, linear solutions are inefficient. The same system control matrix needs to be decomposed into LU values ​​separately in two independent solution processes. This prevents the use of the linear algebra library's ability to batch solve multiple right-hand side terms, and the repeated matrix decomposition operations further increase computation time.

[0006] Third, the utilization rate of hardware computing power is insufficient. The existing solution uses matrix-vector multiplication (GEMV, which belongs to BLAS Level 2 operation) for steps such as backpropagation of field coefficients and calculation of reflection coefficients. Its computation-to-memory access ratio is low, and the computing power is limited by memory bandwidth. It cannot fully activate the SIMD / AVX vectorized parallel hardware throughput of modern CPUs, and the hardware performance cannot be fully utilized.

[0007] Fourth, the computation graph cannot be reused. Common subexpressions across function boundaries cannot be optimized by the compiler or runtime environment cache, memory access bandwidth is saturated by repeated matrix read and write operations, failing to fully utilize the cache hierarchy of modern processors.

[0008] Therefore, how to completely decouple the polarization-independent operator construction process from the polarization-dependent excitation solution process in RCWA calculation, and achieve synchronous parallel solution of dual polarizations while maintaining complete physical accuracy equivalence, is a technical problem that urgently needs to be solved in this field. Summary of the Invention

[0009] In view of this, the present invention provides a polarization calculation optimization method, system, device and medium for RCWA, to solve the problems of highly redundant core steps, low computing power utilization, low solution efficiency and non-reusable calculation graphs in the existing RCWA-based dual polarization calculation.

[0010] This invention provides a polarization calculation optimization method for RCWA, which is applied to the semiconductor under test, wherein the semiconductor under test is pre-divided into an incident layer, a substrate layer and a multilayer grating structure including multiple physical layers using the RCWA method. The method includes: The optical incident parameters and grating geometric parameters of the multilayer grating structure are obtained, and a single environment initialization and operator cascade are performed based on the optical incident parameters and optical geometric parameters to obtain the system control matrix; Construct the incident excitation vectors under the dual polarization states respectively, and then horizontally concatenate the incident excitation vectors under the dual polarization states to obtain the two-column joint excitation matrix; Using the system control matrix, the two right-hand terms of the two columns of the bi-column joint excitation matrix are solved linearly in parallel to obtain the bipolar forward field coefficient matrix; Based on the aforementioned dual-polarization forward field coefficient matrix, the reflection coefficient matrix under the dual-polarization state is obtained by parallel matrix-matrix multiplication. The Jones matrix of the multilayer grating structure is obtained by assembling the reflection coefficient matrix under dual polarization states.

[0011] Optionally, the physical layers of the multilayer grating structure increase sequentially from low to high; Based on the optical incident parameters and optical geometric parameters, a single environmental initialization and operator cascade are performed to obtain the system control matrix, including: The harmonic cutoff number is preset, and the total diffraction order and diffraction order index are determined according to the harmonic cutoff number; Based on the total diffraction order and the diffraction order index, the substrate is constructed according to the optical incident parameters, and the normalized wave vector diagonal matrix pair, azimuth diagonal matrix pair, incident layer feature matrix pair and substrate layer feature matrix pair are calculated respectively. Traverse all physical layers of the multilayer grating structure, perform eigenlayer loops based on the grating geometric parameters, normalized wave vector diagonal matrix pairs, and azimuth diagonal matrix pairs, and construct the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer layer by layer. The substrate is initialized based on the substrate feature matrix pairs to obtain the corresponding substrate boundary matrix pairs. Then, based on the substrate boundary matrix pairs and the single-layer state matrix and single-layer propagation matrix of each physical layer, the process is recursively performed from the substrate layer to the incident layer. Obtain the global boundary matrix pairs corresponding to each physical layer; The system control matrix is ​​obtained by assembling the global boundary matrix pair corresponding to the highest physical layer and the incident layer feature matrix pair.

[0012] Optionally, the optical incident parameters include incident wavelength, incident polar angle, azimuth angle, incident layer refractive index, and substrate layer refractive index; The grating geometry parameters include the thickness and dielectric constant distribution of each physical layer; Traversing all physical layers of the multilayer grating structure, and performing eigenlayer iterations based on the grating geometric parameters, normalized wave vector diagonal matrix pairs, and azimuth diagonal matrix pairs, layer by layer, the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer are constructed, including: Select the l-th physical layer and construct the corresponding Toplitz matrix pair based on the dielectric constant distribution of the selected physical layer; Based on the normalized wave vector diagonal matrix pair and the constructed Topplitz matrix pair, operator matrices are constructed for the TE polarization mode and the TM polarization mode respectively, and the operator matrix pair corresponding to the selected physical layer is obtained. The selected physical layer operator matrix pairs are decomposed into eigenvalues ​​to obtain the corresponding combination of eigenma matrices. Based on the combination of the azimuth diagonal matrix pair and the characteristic matrix of the selected physical layer, the field component coupling matrix is ​​constructed; and the constructed field component coupling matrix is ​​assembled into the single-layer state matrix of the corresponding physical layer. Based on the selected physical layer thickness and the corresponding feature matrix combination, a propagation component matrix is ​​constructed; and the constructed propagation component matrix is ​​assembled into a single-layer propagation matrix of the corresponding physical layer. By traversing each physical layer of the multilayer grating structure, the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer are obtained layer by layer in the same way.

[0013] Optionally, the normalized wave vector diagonal matrix pair includes the x-direction wave vector diagonal matrix and the y-direction wave vector diagonal matrix; Each physical layer’s Toplitz matrix pair includes a dielectric constant matrix and an inverse dielectric constant matrix, and each physical layer’s eigenma matrix combination includes a first eigenvector matrix, a second eigenvector matrix, a first eigenvalue diagonal matrix, and a second eigenvalue diagonal matrix. For the l-th physical layer, the specific calculation formula for the operator matrix pair is as follows: ; ; in, and These are all components of the operator matrix pair of the l-th physical layer, specifically the TE mode operator matrix and the TM mode operator matrix, respectively; P is the inverse permittivity matrix of the l-th physical layer. y is the wave vector diagonal matrix; A and B are components of the intermediate matrix pair constructed based on the wave vector diagonal matrix in the x direction and the dielectric constant matrix of the l-th physical layer, specifically the first intermediate matrix and the second intermediate matrix, respectively. The specific calculation formulas for the first intermediate matrix A and the second intermediate matrix B of the l-th physical layer are as follows: ; ; in, The diagonal matrix of the wave vector in the x-direction. Let l be the dielectric constant matrix of the l-th physical layer. It is the inverse matrix of the dielectric constant matrix of the l-th physical layer; For the l-th physical layer, the specific formula for calculating the feature matrix combination is as follows: ; ; Where M1 and M2 are the first eigenvector matrix and the second eigenvector matrix in the feature matrix combination of the l-th physical layer, respectively, and Q1 and Q2 are the first eigenvalue diagonal matrix and the second eigenvalue diagonal matrix in the feature matrix combination of the l-th physical layer, respectively. For the l-th physical layer, the specific formula for calculating the field component coupling matrix is ​​as follows: ; ; ; ; ; ; ; ; Among them, V ss V sp V ps V pp W ss W sp W ps and W pp All are field components in the field component coupling matrix of the l-th physical layer; F c and F s These are all components of the diagonal matrix pair of the azimuth angles of the l-th physical layer, specifically the cosine direction matrix and the sine direction matrix, respectively; V 11 V 12 V 21 and V 22 All are coupling submatrices of the l-th physical layer; The specific formula for calculating the coupling submatrix of the l-th physical layer is: ; ; ; ; Among them, A -1 and B -1 These are the inverse matrices of the first intermediate matrix A and the second intermediate matrix B of the l-th physical layer, respectively. Each encapsulated physical layer's single-layer state matrix includes a mode coupling matrix and a field coupling matrix. The specific expressions for the mode coupling matrix and field coupling matrix of the l-th physical layer are as follows: ; ; Among them, V l and W l These are the mode coupling matrix and field coupling matrix of the l-th physical layer, respectively; The specific formula for calculating the propagation component matrix of the l-th physical layer is: ; ; Where X1 and X2 are both propagation component matrices of the l-th physical layer, dl Let be the thickness of the l-th physical layer, and k0 be the preset free space wave vector; The specific expression for the single-layer propagation matrix of the l-th physical layer, which is encapsulated, is as follows: ; Among them, X l Let l be the single-layer propagation matrix of the l-th physical layer. Let L be an L×L zero matrix, where L is the total diffraction order.

[0014] Optionally, the substrate boundary matrix pair includes the substrate forward scattering matrix and the substrate backscattering matrix; The specific expression for the substrate boundary matrix pair is: ; ; in, and Y1 and Z2 are the substrate forward scattering and substrate back scattering matrices, respectively, with j being the imaginary unit; Y2 and Z2 are components of the substrate characteristic matrix pair, specifically the substrate admittance matrix and substrate impedance matrix, respectively; I L It is an L×L dimensional identity matrix. Let L be an L×L zero matrix, where L is the total diffraction order; Each physical layer's global boundary matrix pair includes a global forward cumulative scattering matrix and a global backward cumulative scattering matrix; The specific expression for the global boundary matrix of the l-th physical layer is: ; ; in, and V represents the global forward cumulative scattering matrix and the global backward cumulative scattering matrix of the l-th physical layer, respectively. l and W l Let X be the mode coupling matrix and field coupling matrix in the single-layer state matrix of the l-th physical layer, respectively. l Let l be the single-layer propagation matrix of the l-th physical layer. Let I be the recursive coefficient matrix of the l-th physical layer; 2L for An identity matrix of dimensionality; The specific expression of the assembled system control matrix is ​​as follows: ; in, The system control matrix; Y1 and Z1 are components of the incident layer characteristic matrix, specifically the incident layer admittance matrix and the incident layer impedance matrix, respectively; F11 F 12 F 21 and F 22 The global forward cumulative scattering matrix of the l-th physical layer The four obtained after partitioning A submatrix of dimension G 11 G 12 G 21 and G 22 The global backscattering matrix of the l-th physical layer The four obtained after partitioning A submatrix of dimension.

[0015] Optionally, incident excitation vectors under dual polarization states are constructed separately, and the incident excitation vectors under dual polarization states are horizontally concatenated to obtain a two-column joint excitation matrix, including: Define a zero-order diffraction selection vector, and based on the zero-order diffraction selection vector, define the first component field vector in the p-polarization state and the second component field vector in the s-polarization state respectively; Based on the first component field vector, a first single-column excitation vector in the p-polarization state is constructed; and based on the second component field vector, a second single-column excitation vector in the s-polarization state is constructed. The first single-column excitation vector and the second single-column excitation vector are horizontally concatenated along the column direction to obtain the dual-column joint excitation matrix.

[0016] Optionally, both the first component field vector and the second component field vector include four tangential field components; The specific expressions for the four tangential field components in the first component field vector are as follows: ; in, , , and These are the four tangential field components in the first component field vector. Choosing a vector for zero-order diffraction and These are the polar angle of incidence and the refractive index of the incident layer, respectively, in the optical incident parameters; The specific expressions for the four tangential field components in the second component field vector are as follows: ; in, , , and These are the four tangential field components in the second component field vector; The specific expressions for the constructed first and second single-column activation vectors are as follows: ; ; in, and These are the first and second single-column excitation vectors, respectively, both of which are L×1 dimensional column vectors, where L is the total diffraction order; and All of these are components of the incident layer characteristic matrix calculated based on the optical incident parameters, specifically the incident layer admittance matrix and the incident layer impedance matrix; j is the imaginary unit. The specific expression for the concatenated bi-column joint excitation matrix is ​​as follows: ; in, It is a two-column joint activation matrix, specifically a 2L×2 dimensional column vector.

[0017] Optionally, the first component field vector in the p-polarization state specifically includes the first incident layer electric field component. and the first incident layer magnetic field component The second component field vector in the s-polarization state specifically includes the second incident layer electric field component. Second incident layer magnetic field component ; Based on the aforementioned dual-polarization forward field coefficient matrix, the reflection coefficient matrix under dual polarization states is obtained through parallel matrix-matrix multiplication calculations, including: According to the column direction, the first incident layer electric field component in the first component field vector and the second incident layer electric field component in the second component field vector are horizontally spliced ​​together to obtain the first tangential field splicing matrix; and the first incident layer magnetic field component in the first component field vector and the second incident layer magnetic field component in the second component field vector are horizontally spliced ​​together to obtain the second tangential field splicing matrix. Divide the dual-polarization forward field coefficient matrix into rows to obtain the upper half coefficient submatrix and the lower half coefficient submatrix; The first reflection coefficient matrix in the s-polarization state is calculated based on the first tangential field splicing matrix, the upper half-coefficient sub-matrix, the lower half-coefficient sub-matrix, and the global boundary matrix pair of the highest physical layer; and the second reflection coefficient matrix in the p-polarization state is calculated based on the second tangential field splicing matrix, the upper half-coefficient sub-matrix, the lower half-coefficient sub-matrix, the global boundary matrix pair of the highest physical layer, and the incident layer feature matrix pair.

[0018] Optionally, the specific formula for calculating the first reflection coefficient matrix under the s-polarization state is as follows: ; in, Let be the first reflection coefficient matrix under the s-polarization state, specifically an L×2 dimensional vector, where L is the total diffraction order; and These are the upper and lower coefficient submatrices, respectively. The first tangential field splicing matrix is ​​specifically an L×2 dimensional vector; The specific formula for calculating the second reflection coefficient matrix under the p-polarization state is as follows: ; in, This is the second reflection coefficient matrix under p-polarization state. G is the second tangential field splicing matrix, which consists of L×2 dimensional vectors; 11 G 12 G 21 and G 22 The four global backscattering matrices obtained by partitioning the global boundary matrix of the highest physics layer are: A submatrix of dimension j; j is the imaginary unit. It is the inverse of the incident layer impedance matrix in the incident layer characteristic matrix pair.

[0019] Optionally, the Jones matrix of the multilayer grating structure is assembled based on the reflection coefficient matrix under dual polarization states, including: The zero-order component at the Nth position is extracted from the first reflection coefficient matrix and the second reflection coefficient matrix respectively, and the zero-order reflection coefficient of the s component and the zero-order reflection coefficient of the p component under the p polarization state, and the zero-order reflection coefficient of the p component and the zero-order reflection coefficient of the s component under the s polarization state are obtained respectively. Based on the extracted zero-order reflection coefficients of the s component and p component under the p polarization state, as well as the zero-order reflection coefficients of the p component and s component under the s polarization state, the Jones matrix of the multilayer grating structure is assembled. The specific expression of the assembled Jones matrix is ​​as follows: ; in, For the Jones matrix, and These are the zero-order reflection coefficients of the p-component and s-component, respectively, under the p-polarization state. and denoted as the zero-order reflection coefficients of the p-component and s-component respectively under the s-polarization state, and N is the harmonic cutoff number.

[0020] Optionally, using the system control matrix, the two right-hand side terms of the two-column joint excitation matrix are solved linearly in parallel to obtain the dual-polarization forward field coefficient matrix, including: The linear equation solver is invoked to perform a single LU decomposition on the system control matrix to obtain the LU decomposition result. Based on the LU decomposition results, forward and backward permutations are performed in parallel on the two right-hand terms of the bi-column joint excitation matrix to obtain the bipolar forward field coefficient matrix.

[0021] Optionally, the specific expression for the matrix equation used to solve the dual-polarization forward field coefficient matrix is ​​as follows: ; in, The dual-polarization forward field coefficient matrix is ​​specifically a 2L×2 dimensional column vector, where L is the total diffraction order; FG is the system control matrix, and D... mat Let be the bi-column joint excitation matrix.

[0022] Optionally, after obtaining the dual-polarization forward field coefficient matrix, the method further includes: Using matrix-matrix multiplication, based on the global boundary matrix of each physical layer, as well as the single-layer state matrix and single-layer propagation matrix, the dual-polarization forward field coefficient matrix is ​​recursively derived layer by layer from the incident layer to the substrate layer, thereby obtaining the dual-polarization field coefficient matrix of each physical layer.

[0023] Optionally, based on the global boundary matrix of each physical layer, as well as the single-layer state matrix and single-layer propagation matrix, the dual-polarization forward field coefficient matrix is ​​recursively derived layer by layer backward from the incident layer to the substrate layer, including: For the highest physical layer, matrix-matrix multiplication is used to perform backward recursion on the dual-polarization forward field coefficient matrix based on the global boundary matrix, single-layer state matrix, and single-layer propagation matrix of the highest physical layer, to obtain the dual-polarization field coefficient matrix of the highest physical layer. The specific formula for calculating the dual polarization field coefficient matrix of the highest physical layer is as follows: ; in, This is the dual polarization field coefficient matrix of the highest physical layer. Let be the dual-polarization forward field coefficient matrix. These are the components in the single-layer state matrix of the highest physical layer. This is the single-layer propagation matrix of the highest physical layer. This is the recursive coefficient matrix of the highest physical layer. It is the inverse of the global forward cumulative scattering matrix in the global boundary matrix of the highest physical layer; For each current physical layer from the second highest physical layer to the lowest physical layer, matrix-matrix multiplication is used layer by layer. Based on the global boundary matrix, single-layer state matrix and single-layer propagation matrix of the previous physical layer, the dual polarization field coefficient matrix of the previous physical layer is recursively derived to obtain the dual polarization field coefficient matrix of each current physical layer. The specific formula for calculating the dual polarization field coefficient matrix of the l-th physical layer is: ; in, Let be the dual polarization field coefficient matrix of the l-th physical layer. This is the dual polarization field coefficient matrix of the (l+1)th physical layer. These are the components in the single-layer state matrix of the (l+1)th physical layer. This is the single-layer propagation matrix of the (l+1)th physical layer. This is the recursive coefficient matrix of the (l+1)th physical layer. is the inverse of the global forward cumulative scattering matrix in the global boundary matrix pair of the (l+1)th physical layer; l is an integer satisfying 1≤l≤t-1, and t is the total number of physical layers in the multilayer grating structure; For the substrate layer, matrix-matrix multiplication is used to backward recursively derive the dual polarization field coefficient matrix of the lowest physical layer based on the global boundary matrix, single-layer state matrix, and single-layer propagation matrix of the lowest physical layer, so as to obtain the dual polarization field coefficient matrix of the substrate layer. The specific formula for calculating the dual polarization field coefficient matrix of the substrate is as follows: ; in, The matrix represents the dual polarization field coefficients of the substrate. This is the dual polarization field coefficient matrix of the lowest physical layer. These are the components in the single-layer state matrix of the lowest physical layer. This is the single-layer propagation matrix of the lowest physical layer. This is the recursive coefficient matrix of the lowest physical layer. The inverse of the global forward cumulative scattering matrix in the global boundary matrix of the lowest physical layer. In addition, the present invention also provides a polarization calculation optimization system for RCWA, which is applied to the semiconductor under test, wherein the semiconductor under test is pre-divided into an incident layer, a substrate layer and a multilayer grating structure including multiple physical layers using the RCWA method. The system is applied in the aforementioned polarization calculation optimization method for RCWA, including: The parameter preprocessing module is used to obtain the optical incident parameters and grating geometric parameters of the multilayer grating structure, and to perform a single environment initialization and operator cascade based on the optical incident parameters and optical geometric parameters to obtain the system control matrix. The excitation splicing module is used to construct the incident excitation vectors under the dual polarization states respectively, and to horizontally splice the incident excitation vectors under the dual polarization states to obtain a two-column joint excitation matrix; The parallel solution module is used to perform a parallel linear solution on the two right-hand terms of the two-column joint excitation matrix using the system control matrix to obtain the dual polarization forward field coefficient matrix. The matrix calculation module is used to obtain the reflection coefficient matrix under the dual polarization state by matrix-matrix multiplication in parallel calculation based on the dual polarization forward field coefficient matrix; and to assemble the Jones matrix of the multilayer grating structure according to the reflection coefficient matrix under the dual polarization state.

[0024] Furthermore, the present invention also provides a polarization calculation optimization apparatus for RCWA, including a processor, a memory, and a computer program stored in the memory and executable on the processor. When the computer program is executed, it implements the method steps in the aforementioned polarization calculation optimization method for RCWA.

[0025] Furthermore, the present invention also provides a computer storage medium comprising: at least one instruction that, when executed by a computer, implements the method steps in the aforementioned polarization calculation optimization method for RCWA.

[0026] The beneficial effects of this invention are: By obtaining the optical incident parameters and grating geometric parameters of the multilayer grating structure based on the RCWA method, complete known conditions can be provided for subsequent calculations and analyses. Performing a single environment initialization and operator cascading based on the optical incident parameters and grating geometric parameters enables the one-time construction of the system control matrix, avoiding the redundant operations of repeated environment initialization and operator cascading in traditional polarization-separation calculations, thus reducing repetitive computations from the computational source. Simultaneously, the constructed system control matrix is ​​polarization-independent, satisfying the requirement of one decomposition and multiple solutions. Based on this system control matrix, subsequent linear solutions can be performed in parallel on the two right-hand sides of the two-column joint excitation matrix to obtain the dual-polarization forward field coefficient matrix, without needing to separately solve for the two polarization states. The system control matrix construction and decomposition operations significantly reduce computational redundancy. Furthermore, when constructing the bi-column joint excitation matrix, only a simple horizontal concatenation of the incident excitation vectors under the dual polarization states is required to achieve parallel computation of dual polarization, without the need for additional complex dimension expansion operations, thus greatly improving computational efficiency. After calculating the dual polarization forward field coefficient matrix, the reflection coefficient matrix under the dual polarization states is obtained through matrix-matrix parallel computation based on the dual polarization forward field coefficient matrix. Then, the Jones matrix used to describe the polarization characteristics of the multilayer grating is assembled, transforming the original two independent vector-matrix operations into a single matrix-matrix operation. This fully utilizes the parallel computing capabilities of modern computing hardware, further compressing the overall computation time. The present invention provides a polarization calculation optimization method, system, device, and medium for RCWA. By constructing a polarization-independent system control matrix in one step and combining it with a design for parallel solution of dual polarization excitations, it eliminates the redundant steps of repeatedly constructing and decomposing the system matrix during polarization-specific calculations. It also fully leverages the parallel computing performance of modern hardware, significantly improving the overall efficiency of polarization calculations while ensuring the physical robustness of the algorithm and the consistency of calculation accuracy. At the same time, it achieves a balance between memory usage and computing power consumption, is adaptable to computing devices with different hardware configurations, and can meet the computational needs of polarization-related optical measurements in the semiconductor field. Attached Figure Description

[0027] The features and advantages of the invention will be more clearly understood by referring to the accompanying drawings, which are schematic and should not be construed as limiting the invention in any way. In the drawings: Figure 1 A flowchart of a polarization calculation optimization method for RCWA according to Embodiment 1 of the present invention is shown; Figure 2 The simulation results of the trapezoidal multilayer grating structure simulated using a complete polarization optimization calculation method in Embodiment 1 of the present invention are shown. Figure 3 A structural diagram of a polarization calculation optimization system for RCWA according to Embodiment 2 of the present invention is shown. Detailed Implementation

[0028] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0029] In this embodiment of the invention, the term "and / or" describes the relationship between associated 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 character " / " generally indicates that the preceding and following associated objects have an "or" relationship.

[0030] It should be noted that the terms "first," "second," etc., in the specification, claims, and drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence.

[0031] In this embodiment of the invention, the term "multiple" refers to two or more, and other quantifiers are similar.

[0032] Example 1 A polarization calculation optimization method for RCWA is applied to a semiconductor under test, wherein the semiconductor under test is pre-divided into an incident layer, a substrate layer and a multilayer grating structure including multiple physical layers using the RCWA method. like Figure 1 As shown, the method includes: S1: Obtain the optical incident parameters and grating geometric parameters of the multilayer grating structure, and perform a single environment initialization and operator cascade based on the optical incident parameters and optical geometric parameters to obtain the system control matrix; S2: Construct the incident excitation vectors under the dual polarization states respectively, and horizontally concatenate the incident excitation vectors under the dual polarization states to obtain the two-column joint excitation matrix; S3: Using the system control matrix, perform parallel linear solutions on the two right-hand terms of the two-column joint excitation matrix to obtain the dual-polarization forward field coefficient matrix; S4: Based on the aforementioned dual-polarization forward field coefficient matrix, the reflection coefficient matrix under the dual-polarization state is obtained by parallel matrix-matrix multiplication. S5: Based on the reflection coefficient matrix under dual polarization states, the Jones matrix of the multilayer grating structure is assembled.

[0033] In this embodiment, by obtaining the optical incident parameters and grating geometric parameters of the multilayer grating structure based on the RCWA method, complete known conditions can be provided for subsequent calculations and analyses. Performing a single environment initialization and operator cascading based on the optical incident parameters and grating geometric parameters enables the one-time construction of the system control matrix, avoiding the redundant operations of repeatedly performing environment initialization and operator cascading in traditional polarization-separation calculations, thus reducing repetitive computations from the computational source. Simultaneously, the constructed system control matrix is ​​polarization-independent, satisfying the requirement of one decomposition and multiple solutions. Based on this system control matrix, subsequent linear solutions can be performed in parallel on the two right-hand sides of the two-column joint excitation matrix to obtain the dual-polarization forward field coefficient matrix, without needing to perform linear solutions on the two polarizations. The system control matrix construction and decomposition operations are performed separately for each state, significantly reducing computational redundancy. Furthermore, when constructing the dual-column joint excitation matrix, only a simple horizontal concatenation of the incident excitation vectors under the dual polarization states is required to achieve parallel computation of dual polarization, without the need for additional complex dimension expansion operations, thus greatly improving computational efficiency. After calculating the dual polarization forward field coefficient matrix, the reflection coefficient matrix under the dual polarization states is obtained through matrix-matrix parallel computation based on the dual polarization forward field coefficient matrix, and then assembled to obtain the Jones matrix used to describe the polarization characteristics of the multilayer grating. This transforms the original two independent vector-matrix operations into a single matrix-matrix operation, which can fully utilize the parallel computing capabilities of modern computing hardware and further compress the overall computation time.

[0034] The polarization calculation optimization method for RCWA in this embodiment constructs a polarization-independent system control matrix in one step and combines it with a design for parallel solution of dual polarization excitations. This eliminates the redundant steps of repeatedly constructing and decomposing the system matrix during polarization-specific calculations, and fully leverages the parallel computing performance of modern hardware. While ensuring the physical robustness of the algorithm and the consistency of calculation accuracy, it greatly improves the overall efficiency of polarization calculations. At the same time, it achieves a balance between memory usage and computing power consumption, is adaptable to computing devices with different hardware configurations, and can meet the calculation needs of polarization-related optical measurements in the semiconductor field.

[0035] The following section further explains the various steps of the polarization calculation optimization method for RCWA in this embodiment.

[0036] For the rigorous coupled-wave analysis method (RCWA), in the specific analysis process, the semiconductor structure under test is divided into three parts along the depth direction: the incident layer, the multilayer grating structure, and the substrate layer. The multilayer grating structure is divided into multiple physical layers according to the material stacking. Each physical layer has corresponding geometric parameters such as refractive index, thickness, and grating period.

[0037] Before single-stage environment initialization and operator cascading, a rigorous coupled-wave analysis method is first employed for structural partitioning. This method clearly delineates the semiconductor under test along its depth direction, determines the arrangement order of each physical layer, and identifies optical incident parameters such as material refractive index and grating geometric parameters such as thickness for each physical layer, providing a clear structural foundation for subsequent calculations. After structural partitioning, when performing single-stage environment initialization and operator cascading, only one polarization-independent system control matrix needs to be constructed based on the parameters of all physical layers. This eliminates the need for separate constructions for TE and TM polarization states, thus eliminating redundant operations from the initial calculation stage.

[0038] Specifically, the physical layers of the multilayer grating structure increase sequentially from low to high. Based on this multilayer grating structure, the optical incident parameters include incident wavelength, incident polar angle, azimuth angle, incident layer refractive index, and substrate layer refractive index; the grating geometric parameters include the thickness and dielectric constant distribution of each physical layer.

[0039] By using the aforementioned optical incident parameters, the incident conditions of the incident light and the basic optical properties of each layer can be clearly defined. Combined with the thickness and dielectric distribution of each physical layer described by the aforementioned grating geometric parameters, all the computational input information of the multilayer grating structure can be completely determined without adjusting parameters for different polarizations, thus ensuring the polarization independence of the system control matrix and providing a foundation for subsequent single-builds and reuse.

[0040] Preferably, in step S1, based on the optical incident parameters and optical geometric parameters, a single environmental initialization and operator cascade are performed to obtain the system control matrix, including: S11: Preset the harmonic cutoff number, and determine the total diffraction order and diffraction order index according to the harmonic cutoff number; S12: Based on the total diffraction order and the diffraction order index, the substrate is constructed according to the optical incident parameters, and the normalized wave vector diagonal matrix pair, azimuth diagonal matrix pair, incident layer feature matrix pair and substrate layer feature matrix pair are calculated respectively. S13: Traverse all physical layers of the multilayer grating structure, perform eigenlayer looping based on the grating geometric parameters, normalized wave vector diagonal matrix pairs and azimuth diagonal matrix pairs, and construct the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer layer by layer. S14: Initialize the substrate layer based on the substrate layer characteristic matrix pairs to obtain the corresponding substrate boundary matrix pairs. Then, based on the substrate boundary matrix pairs and the single-layer state matrix and single-layer propagation matrix of each physical layer, perform a layer-by-layer recursion from the substrate layer to the incident layer. Obtain the global boundary matrix pairs corresponding to each physical layer; S15: The system control matrix is ​​assembled based on the global boundary matrix pair corresponding to the highest physical layer and the incident layer feature matrix pair.

[0041] First, the harmonic cutoff number is set. This number represents the number of diffraction orders retained in the RCWA calculation. The larger the cutoff number, the more diffraction orders participate in the calculation, and the higher the calculation accuracy. This parameter can be flexibly adjusted according to the actual accuracy requirements to adapt to the calculation requirements of different scenarios. After determining the harmonic cutoff number, the corresponding total diffraction order and diffraction order index are obtained, providing a clear basis for the dimension and arrangement rules of subsequent matrices. Then, based on the total diffraction order and diffraction order index, combined with the known optical incident parameters, the substrate is constructed, and each basic diagonal matrix and eigenvalue matrix pair is calculated sequentially, completing the initialization preparation of the entire calculation environment. These matrices are only related to the incident conditions and material properties, and do not involve polarization-related parameters, so they will not change with the polarization state. Then, all physical layers in the multilayer grating structure are traversed. Combining the grating geometry parameters of each physical layer with the obtained basic matrices, the intrinsic calculations are performed on each physical layer one by one to obtain the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer, completing the basic preparation for operator cascading. After all the single-layer matrices of all physical layers are constructed, starting from the substrate layer, the substrate boundary matrix pair is first initialized. Then, the process is recursively applied from the substrate layer to the incident layer. Each layer is combined with the global boundary matrix pair obtained from the previous layer, as well as the single-layer state matrix and single-layer propagation matrix of the current layer, to calculate the corresponding global boundary matrix pair for the current layer. This process continues until the highest physical layer is reached. Finally, the global boundary matrix pair of the highest physical layer is combined with the incident layer feature matrix pair to assemble the complete system control matrix. The entire process only needs to be executed once and does not require repeated calculations.

[0042] Specifically, in step S11, if the harmonic cutoff number is N, then the total number of diffraction orders is L=2N+1. The diffraction order indices are arranged in integer order of [-N, N], and a unique index is assigned to each diffraction order to facilitate the location and calculation of matrix elements.

[0043] Specifically, in step S12, the normalized wave vector diagonal matrix pair includes the normalized wave vector diagonal matrices for TE polarization mode and TM polarization mode. These two diagonal matrices are determined only by the incident conditions and diffraction order, do not change with the grating geometric details, and are independent of the subsequent solution process. They can be used throughout the entire process after a single calculation. Similarly, the azimuth diagonal matrix pair, the incident layer characteristic matrix pair, and the substrate layer characteristic matrix pair are calculated without involving polarization-related specific adjustments. All the fundamental matrices obtained can be used for the subsequent construction of polarization-independent system control matrices.

[0044] For a normalized wave vector diagonal matrix pair, it includes the x-direction wave vector diagonal matrix. and the diagonal matrix of the wave vector in the y direction Defined at a given operating wavelength The wave vector in free space (i.e., the free space wave vector) is Let the incident polar angle be... azimuth angle is The grating period is Then its reciprocal lattice vector size is .

[0045] At a given operating wavelength Below, the incident wave vector is and The directional components are defined as follows: , ; in, Let be the component of the incident wave vector in the x-direction. Let be the component of the incident wave vector in the y-direction. The refractive index of the incident layer (such as air).

[0046] For the m-th order diffraction, its The y-direction wave vector components and the y-direction wave vector components are respectively: ; ; Where, k x,m Let k be the wave vector component in the x-direction of the semiconductor under test at the m-th diffraction order. y,m Let y be the wave vector component of the semiconductor under test in the m-th diffraction order; Define the normalized wave vector component: ; ; in, and These are the normalized x-direction wave vector component (i.e., the normalized x-direction wave vector) and the y-direction wave vector component (i.e., the normalized y-direction wave vector), respectively. Based on the above normalized wave vector components, construct the diagonal matrix: ; ; in, and These are the diagonal matrices of the wave vectors in the x and y directions, respectively. To perform operations on diagonal elements, The x-direction wave vector components are the normalized components at each diffraction order. For the normalized y-direction wave vector component, IL Let L be the L×L identity matrix, where L is the total diffraction order.

[0047] For the azimuth diagonal matrix pair, the incident layer characteristic matrix pair, and the substrate layer characteristic matrix pair, the azimuth diagonal matrix pair includes the cosine direction matrix F. c and the sinusoidal direction matrix F s The incident layer feature matrix pair includes the incident layer feature admittance matrix. and the characteristic impedance matrix of the incident layer The substrate feature matrix includes the substrate feature admittance matrix. and substrate characteristic impedance matrix .

[0048] For the incident layer (also known as the incident space), its refractive index ) and substrate layer (also known as substrate space, whose refractive index) ), calculate the diffraction order separately. Directional wave vector components: ; The value of the square root satisfies the radiation condition: if If l is 1, then a negative sign is taken to ensure physical attenuation behavior. Let l be the wave vector component of the incident layer in the z-direction of the m-th diffraction order; when l is 2... The z-direction wave vector component of the substrate layer in the m-th diffraction order is denoted as .

[0049] Define the azimuth angle of each diffraction order: ; in, The azimuth angle of the m-th diffraction order; Therefore, the corresponding cosine direction matrix and sine direction matrix are constructed: ; ; in, and These are the cosine direction matrix and the sine direction matrix, respectively. For each diffraction order, the azimuth angle is given. And the characteristic admittance matrix and characteristic impedance matrix: ; ; in, and These are the incident layer admittance matrix and the incident layer impedance matrix, respectively. and These are the substrate admittance matrix and the substrate impedance matrix, respectively; both the incident layer admittance matrix and the substrate admittance matrix are characteristic admittance matrices, and both the incident layer impedance matrix and the substrate impedance matrix are characteristic impedance matrices.

[0050] Specifically, in step S13, when performing eigenvalue calculations on any physical layer, a Fourier expansion is first performed based on the dielectric constant distribution of that physical layer to obtain the dielectric constant matrix. Then, combined with the normalized wave vector diagonal matrix and azimuth diagonal matrix, the eigenvalue problems corresponding to the TE and TM modes are constructed respectively. The eigenvector matrix and eigenvalue diagonal matrix corresponding to that physical layer are solved, and then the single-layer state matrix and single-layer propagation matrix are assembled accordingly. The entire calculation process only focuses on the optical and geometric properties of the physical layer itself and does not involve subsequent incident excitation solutions. Therefore, all results are polarization-independent and can be directly used for subsequent parallel calculations of dual polarization without needing to repeatedly calculate eigenvalue results for different polarizations. Furthermore, in steps S14 and S15, when recursively obtaining the global boundary matrix pairs and finally assembling the system control matrix, the entire process is also based on the polarization-independent matrices that have already been calculated for each physical layer. Therefore, the final system control matrix also does not depend on the polarization state and only needs to be calculated once for subsequent parallel solutions of dual polarization.

[0051] Preferably, step S13 includes: S131: Select the l-th physical layer and construct the corresponding Toplitz matrix pair based on the dielectric constant distribution of the selected physical layer; S132: Based on the normalized wave vector diagonal matrix pair and the constructed Topplitz matrix pair, construct operator matrices for the TE polarization mode and the TM polarization mode respectively, and obtain the operator matrix pair corresponding to the selected physical layer. S133: Perform eigenvalue decomposition on the selected physical layer operator matrix pairs to obtain the corresponding eigenma matrix combination; S134: Construct the field component coupling matrix based on the combination of the azimuth diagonal matrix pair and the characteristic matrix of the selected physical layer; and assemble the constructed field component coupling matrix into the single-layer state matrix of the corresponding physical layer; S135: Construct the propagation component matrix based on the selected physical layer thickness and the corresponding feature matrix combination; and assemble the constructed propagation component matrix into a single-layer propagation matrix of the corresponding physical layer; S136: Traverse each physical layer of the multilayer grating structure and obtain the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer in the same way.

[0052] During the eigenlayer loop of any physical layer, the corresponding dielectric constant Fourier expansion matrix is ​​first constructed based on the dielectric constant distribution of that physical layer. This matrix is ​​then rearranged for different grating period directions to obtain Topplitz matrix pairs. The Topplitz matrix accurately organizes the Fourier components of the dielectric distribution, adapting to the dimensionality requirements of subsequent operator construction. Subsequently, combined with the obtained normalized wave vector diagonal matrix pairs, corresponding eigenoperator matrices are constructed for both TE and TM polarization modes. The operators for both modes share the fundamental matrix of the dielectric distribution, differing only in the construction rules of the core terms; all fundamental parameters do not need to be recalculated. After constructing the operator matrix pairs, eigenvalue decomposition is performed on both operators to obtain the eigenvector matrix and eigenvalue diagonal matrix corresponding to each mode, which together form the eigenmatrix combination of that physical layer. Next, using the azimuth diagonal matrix pairs calculated in the preceding steps, combined with the already obtained eigenma matrix combinations, the field component coupling matrices for TE and TM modes are constructed. Then, the field component coupling matrices of the two modes are assembled according to rules to obtain a complete single-layer state matrix. This matrix integrates the eigenvalues ​​of both polarization modes and can be directly used for subsequent recursive calculations without further splitting and adjustment. Afterward, based on the thickness of the current physical layer and the eigenvalues ​​in the eigenma matrix combination, the propagation component matrix is ​​calculated, and the corresponding single-layer propagation matrix is ​​assembled, also integrating the propagation information of dual polarization. After completing the calculation for the current physical layer, the process continues to traverse the next physical layer, repeating the above steps until the corresponding single-layer state matrix and single-layer propagation matrix have been calculated for all physical layers, completing all calculations for the eigenlayer loop and providing complete basic data for subsequent recursive assembly.

[0053] Specifically, the dielectric constant distribution of each physical layer includes the ridge region refractive index, the groove region refractive index, and the duty cycle. Each physical layer's Toplitz matrix pair includes a dielectric constant matrix and an inverse dielectric constant matrix, and each physical layer's eigenma matrix combination includes a first eigenvector matrix, a second eigenvector matrix, a first eigenvalue diagonal matrix, and a second eigenvalue diagonal matrix.

[0054] For the l-th physical layer, step S131 includes: S1311: Calculate the corresponding ridge region dielectric constant based on the ridge region refractive index of the l-th physical layer, and calculate the corresponding trench region dielectric constant based on the trench region refractive index of the l-th physical layer. For each physical layer, its spatially distributed dielectric constant It needs to be expanded using Fourier series. Let the refractive index of the ridge region of the l-th physical layer be... The refractive index of the groove region is Duty cycle is The formulas for calculating the dielectric constants of the ridge region and the slot region are as follows: , .

[0055] S1312: Using the duty cycle, ridge dielectric constant, and trench dielectric constant of the l-th physical layer, perform a Fourier series expansion to calculate the Fourier coefficients corresponding to the dielectric constant of the l-th physical layer and the Fourier coefficients corresponding to the reciprocal of the dielectric constant. The specific formula for calculating the Fourier coefficient of the dielectric constant of the l-th physical layer at the h-th level is: ; in, Let be the Fourier coefficient of the dielectric constant of the l-th physical layer at the h-th order. It is a Fourier series. and These are the ridge and trench dielectric constants of the l-th physical layer, respectively. This represents the duty cycle of the l-th physical layer. The specific formula for calculating the reciprocal of the dielectric constant of the l-th physical layer at the h-th level is: ; in, The Fourier coefficient of the reciprocal of the dielectric constant of the l-th physical layer at the h-th order is given by the following formula: S1313: Based on the Fourier coefficients corresponding to the dielectric constant of the l-th physical layer and the Fourier coefficients corresponding to the reciprocal of the dielectric constant, construct the dielectric constant matrix and the inverse dielectric constant matrix of the corresponding physical layer; The expressions for the dielectric constant matrix and inverse dielectric constant matrix of the l-th physical layer are as follows: ; ; Among them, E i,p Let P be the element in the i-th row and p-th column of the dielectric constant matrix of the l-th slice. i,p Let be the element in the i-th row and p-th column of the inverse permittivity matrix of the l-th slice. Let be the Fourier coefficient of the dielectric constant of the l-th slice at the ip-th order. The Fourier coefficient of the reciprocal of the dielectric constant of the l-th slice is the Fourier coefficient at the ip-th order.

[0056] In the dielectric constant matrix E mentioned above, its first column is... The first line Inverse matrix of dielectric constant Similarly, this embodiment uses Li's rule to apply the dielectric constant. and the reciprocal of the dielectric constant Performing independent Fourier expansions to ensure computational convergence effectively avoids the numerical divergence problem that easily occurs when the grating structure has a large refractive index difference, which is common when the dielectric constant is directly expanded in traditional methods. This improves the stability and accuracy of intrinsic calculations.

[0057] Furthermore, in step S132, within each physical layer, Maxwell's equations are transformed into a system of second-order differential equations. Therefore, in step S132, an intermediate matrix pair is first defined, which includes a first intermediate matrix A and a second intermediate matrix B. The specific calculation formulas for the first intermediate matrix A and the second intermediate matrix B of the l-th physical layer are as follows: ; ; in, The diagonal matrix of the wave vector in the x-direction. Let l be the dielectric constant matrix of the l-th physical layer. It is the inverse matrix of the dielectric constant matrix of the l-th physical layer.

[0058] Based on the first and second intermediate matrices mentioned above, the specific calculation formula for the feature matrix combination for the l-th physical layer is as follows: ; ; Where M1 and M2 are the first and second eigenvector matrices in the feature matrix combination of the l-th physical layer, respectively, and Q1 and Q2 are the first and second eigenvalue diagonal matrices in the feature matrix combination of the l-th physical layer, respectively.

[0059] In the above combination of eigenvector matrices, the first eigenvector matrix M1 and the second eigenvector matrix M2 represent the field distribution coupling coefficients of each spatial harmonic; the first eigenvalue diagonal matrix Q1 and the second eigenvalue diagonal matrix Q2, whose elements correspond to each mode in The propagation constant in the direction.

[0060] Furthermore, in step S134, in order to calculate the field component coupling matrix of each physical layer, it is necessary to first define the coupling sub-matrix of each physical layer, which includes four sub-matrixes.

[0061] The specific formula for calculating the coupling submatrix of the l-th physical layer is: ; ; ; ; Among them, A -1 and B-1 These are the inverse matrices of the first intermediate matrix A and the second intermediate matrix B of the l-th physical layer, respectively. The above coupling sub-matrices represent the coupling relationship between different field components in TE mode and TM mode. The four coupling sub-matrices are combined to obtain the complete field component coupling matrix.

[0062] For the l-th physical layer, the specific formula for calculating the field component coupling matrix is ​​as follows: ; ; ; ; ; ; ; ; Among them, V ss V sp V ps V pp W ss W sp W ps and W pp All are field components in the field component coupling matrix of the l-th physical layer; F c and F s These are all components of the diagonal matrix pair of the azimuth angles of the l-th physical layer, specifically the cosine direction matrix and the sine direction matrix, respectively; V 11 V 12 V 21 and V 22 All are coupling submatrices of the l-th physical layer.

[0063] In step S134, the single-layer state matrix of each encapsulated physical layer includes a mode coupling matrix and a field coupling matrix. The specific expressions for the mode coupling matrix and field coupling matrix of the l-th physical layer are as follows: ; ; Among them, V l and W l These are the mode coupling matrix and field coupling matrix of the l-th physical layer, respectively, both of which are... A 3D matrix.

[0064] In step S135, the specific calculation formula for the propagation component matrix of the l-th physical layer is as follows: ; ; Where X1 and X2 are both propagation component matrices of the l-th physical layer, d l Let be the thickness of the l-th physical layer, and k0 be the preset free space wave vector; The specific expression for the single-layer propagation matrix of the l-th physical layer, which is encapsulated, is as follows: ; Among them, X l Let l be the single-layer propagation matrix of the l-th physical layer, which is also l. A 3D matrix; Let L be an L×L zero matrix, where L is the total diffraction order.

[0065] In step S136, each physical layer constructs a single-layer mode coupling matrix, a single-layer field coupling matrix, and a single-layer propagation matrix according to the above method. The specific details of other physical layers will not be elaborated here.

[0066] Preferably, in step S14, the substrate boundary matrix pair includes the substrate forward scattering matrix and the substrate backscattering matrix; The specific expression for the substrate boundary matrix pair is: ; ; in, and Y1 and Z2 are the substrate forward scattering and substrate back scattering matrices, respectively, with j being the imaginary unit; Y2 and Z2 are components of the substrate characteristic matrix pair, specifically the substrate admittance matrix and substrate impedance matrix, respectively; I L It is an L×L dimensional identity matrix. Let L be an L×L zero matrix, where L is the total diffraction order.

[0067] To derive the global boundary matrix of each physical layer stepwise from the substrate boundary matrix, it is necessary to first define the intermediate coupling matrix of each physical layer, and then calculate the recursive coefficient matrix of each physical layer by combining the single-layer state matrix and single-layer propagation matrix of the corresponding physical layer. Through this step, a bottom-up, physical-layer recursive calculation can be achieved to gradually obtain the global scattering matrix of the entire grating structure.

[0068] Specifically, the formula for calculating the intermediate coupling matrix of the l-th physical layer is as follows: ; Among them, C l F is the intermediate coupling matrix of the l-th physical layer; curr and G currThese are all components of the global boundary matrix of the (l-1)th physical layer, specifically the global forward cumulative scattering matrix and the global backward cumulative scattering matrix of the (l-1)th physical layer; The global forward cumulative scattering matrix F curr The inverse matrix.

[0069] Therefore, the specific formula for calculating the recursive coefficient matrix of the l-th physical layer is as follows: ; Among them, a l Let be the recursive coefficient matrix of the l-th physical layer.

[0070] The aforementioned recursive coefficient matrix represents the increment of the scattered field, which can accurately capture the coupling changes of the scattered field between the current physical layer and the previous recursive process, providing an accurate basis for the incremental correction of the subsequent scattering matrix update, and ensuring the continuity and accuracy of the scattering state transmission during the layer-by-layer recursive process.

[0071] In step S14, each physical layer's global boundary matrix pair includes a global forward cumulative scattering matrix and a global backward cumulative scattering matrix; Based on the above recursive coefficient matrix, the specific expression for the global boundary matrix of the l-th physical layer is: ; ; in, and V represents the global forward cumulative scattering matrix and the global backward cumulative scattering matrix of the l-th physical layer, respectively. l and W l Let X be the mode coupling matrix and field coupling matrix in the single-layer state matrix of the l-th physical layer, respectively. l Let l be the single-layer propagation matrix of the l-th physical layer. Let I be the recursive coefficient matrix of the l-th physical layer; 2L for An identity matrix of dimension 1.

[0072] For the highest physics layer, its global forward cumulative scattering matrix and global backward cumulative scattering matrix are respectively expressed as: and .

[0073] Furthermore, in step S15, before assembling the system control matrix, the global boundary matrix pair of the highest physical layer is first processed into blocks to obtain block boundary matrix pairs. The specific expression for the block boundary matrix pairs obtained after block processing is: ; in, and These are all components of the block boundary matrix pair, specifically the forward block state matrix and the backward block state matrix; F 11 F 12 F 21 and F 22 All are submatrices in the forward block state matrix, G 11 G 12 G 21 and G 22 All of these are submatrices within the backward block state matrix.

[0074] The above F 11 F 12 F 21 and F 22 and G 11 G 12 G 21 and G 22 All A 3D matrix.

[0075] Furthermore, the specific expression of the assembled system control matrix is ​​as follows: ; in, The system control matrix; Y1 and Z1 are components of the incident layer characteristic matrix, specifically the incident layer admittance matrix and the incident layer impedance matrix, respectively; F 11 F 12 F 21 and F 22 The global forward cumulative scattering matrix of the l-th physical layer The four obtained after partitioning A submatrix of dimension G 11 G 12 G 21 and G 22 The global backscattering matrix of the l-th physical layer The four obtained after partitioning A submatrix of dimension.

[0076] Preferably, step S2 in this embodiment includes: S21: Define the zero-order diffraction selection vector, and define the first component field vector in the p-polarization state and the second component field vector in the s-polarization state based on the zero-order diffraction selection vector. S22: Based on the first component field vector, construct the first single-column excitation vector in the p-polarization state; and based on the second component field vector, construct the second single-column excitation vector in the s-polarization state. S23: Horizontally concatenate the first single-column excitation vector and the second single-column excitation vector along the column direction to obtain the dual-column joint excitation matrix.

[0077] The zero-order diffraction selection vector is an L-dimensional column vector that retains only the positions corresponding to the zero-order diffraction order as 1, while the positions corresponding to all other higher-order diffraction orders are all 0. This vector can accurately select the zero-order diffraction results to be extracted, directly isolating redundant calculations of all higher-order diffraction orders. For incident light, only the zero-order diffraction carries the necessary polarization-dependent outgoing light information. Therefore, with this selection vector, it is unnecessary to calculate the field intensity results of all diffraction orders before extracting the zero-order components; instead, the zero-order results are directly located in the final calculation, eliminating unnecessary higher-order term operations from the source of calculation and reducing the overall computational dimensionality. In this embodiment, by defining the zero-order diffraction selection vector and then defining the first component field vector in the p-polarization state and the second component field vector in the s-polarization state, it is convenient to extract the first single-column excitation vector in the p-polarization state and the second single-column excitation vector in the s-polarization state respectively. This facilitates the horizontal concatenation to obtain a two-column joint excitation matrix, allowing for the simultaneous acquisition of zero-order diffraction results in both polarization states in a single solution, eliminating the repetitive steps of two independent calculations and further improving computational efficiency.

[0078] Specifically, in step S21, the zero-order diffraction selection vector is defined. ( ), its first One element is 1, and the rest are 0. For P-polarization ( ) and S-polarization ( The excitation vector is constructed using two incident conditions (i.e., the preset p-polarization incident condition and the preset s-polarization incident condition).

[0079] For P-polarized incident ( (TM mode), defining the first component field vector, which includes four tangential field vectors: ; For S-polarized incident ( (TE mode), defines the second component field vector, which also includes four tangential field vectors: ; in, , , and These are the four tangential field components in the first component field vector. Choosing a vector for zero-order diffraction and These are the polar angle of incidence and the refractive index of the incident layer, respectively, in the optical incident parameters; , , and These are the four tangential field components in the second component field vector.

[0080] Specifically, in step S22, based on the first and second component field vectors mentioned above, a first single-column excitation vector in the p-polarization state and a second single-column excitation vector in the s-polarization state are constructed respectively, and their specific expressions are as follows: ; ; in, and These are the first and second single-column excitation vectors, respectively, both of which are L×1 dimensional column vectors, where L is the total diffraction order; and All of these are components of the incident layer characteristic matrix calculated based on the optical incident parameters, specifically the incident layer admittance matrix and the incident layer impedance matrix; j is the imaginary unit.

[0081] Specifically, in step S23, the specific expression of the concatenated bi-column joint excitation matrix is ​​as follows: ; in, It is a two-column joint activation matrix, specifically a 2L×2 dimensional column vector.

[0082] Preferably, step S3 includes: S31: Call the linear equation solver to perform a single LU decomposition on the system control matrix and obtain the LU decomposition result; S32: Based on the LU decomposition results, perform forward and backward permutations in parallel on the two right-hand terms of the two columns of the bi-column joint excitation matrix to obtain the bipolar forward field coefficient matrix.

[0083] After obtaining the system control matrix and the bi-column joint excitation matrix, a single LU decomposition of the system control matrix is ​​performed by calling the linear equation solver. The LU decomposition results (including the lower triangular matrix and the upper triangular matrix) can be reused to simultaneously perform permutation and solution on the two right-hand side terms of the bi-column joint excitation matrix. This eliminates the need to perform two separate LU decomposition and solution processes for p-polarization and s-polarization, further reducing redundant calculation steps. While retaining the numerical stability advantage of LU decomposition, the calculation results of a single decomposition are maximized, and the solution calculations for the two polarization states are combined into one, effectively compressing the overall calculation time.

[0084] Specifically, in step S31, the linear equation solver refers to a general-purpose linear solution module implemented based on direct LU decomposition or sparse LU decomposition algorithms. It can adapt to system control matrices of different dimensions, ensuring numerical stability and computational efficiency in the solution process. For large-scale diffraction order calculation scenarios, it can further reduce memory usage through sparse storage strategies, adapting to computational needs under different computing power conditions. In this embodiment, the linear equation solver is specifically `solve(FG,D_mat)`, supported by LU decomposition and trigonometric solving algorithms provided by its underlying LAPACK (a linear algebra software package, a general-purpose subroutine library for numerical linear algebra calculations). It can directly perform a single LU decomposition on the system control using mature numerical computation tools, ensuring solution accuracy and robustness. Subsequently, forward and backward permutations are performed in parallel on the two right-hand sides of the bi-column joint excitation matrix, directly obtaining an output matrix of 2L×2 dual-polarization forward field coefficients. The first column of this matrix corresponds to the zeroth-order diffraction field coefficients for p-polarized incident radiation, and the second column corresponds to the zeroth-order diffraction field coefficients for s-polarized incident radiation.

[0085] Specifically, in step S32, the matrix equation used to solve the dual-polarization forward field coefficient matrix is ​​expressed as follows: ; in, The dual-polarization forward field coefficient matrix is ​​specifically a 2L×2 dimensional column vector, where L is the total diffraction order; FG is the system control matrix, and D... mat Let be the bi-column joint excitation matrix.

[0086] The aforementioned matrix equation reflects the linear correspondence between the incident field, the structure's scattering characteristics, and the exit field. By integrating the solutions for the two polarization states into a single matrix equation, the independence of the calculation process under different polarizations is ensured, while the reuse of decomposition steps is achieved. This reduces redundant repetitive calculations at the algorithm level, fundamentally improving the overall efficiency of polarization calculation. The form of this matrix equation does not require any format conversion of the system control matrix; it directly adopts the original recursive framework of the RCWA algorithm. Polarization calculation optimization can be achieved without modifying the core recursive logic, demonstrating good compatibility and ease of integration into existing RCWA implementations.

[0087] Preferably, step S4 includes: S41: According to the column direction, the first incident layer electric field component in the first component field vector and the second incident layer electric field component in the second component field vector are horizontally spliced ​​together to obtain the first tangential field splicing matrix; and the first incident layer magnetic field component in the first component field vector and the second incident layer magnetic field component in the second component field vector are horizontally spliced ​​together to obtain the second tangential field splicing matrix. S42: Divide the dual-polarization forward field coefficient matrix into rows to obtain the upper half coefficient submatrix and the lower half coefficient submatrix; S43: Calculate the first reflection coefficient matrix in the s-polarization state based on the first tangential field splicing matrix, the upper half-coefficient sub-matrix, the lower half-coefficient sub-matrix, and the global boundary matrix pair of the highest physical layer; and calculate the second reflection coefficient matrix in the p-polarization state based on the second tangential field splicing matrix, the upper half-coefficient sub-matrix, the lower half-coefficient sub-matrix, the global boundary matrix pair of the highest physical layer, and the incident layer feature matrix pair.

[0088] When the aforementioned steps output a 2L×2 dual-polarization forward field coefficient matrix, dividing this matrix into rows yields two L×2 submatrices: an upper submatrix and a lower submatrix. These two submatrices reflect the coefficient distribution characteristics of the forward diffraction field under different polarizations, providing a foundation for subsequent polarization-specific reflection coefficient extraction. The first and second tangential field concatenation matrices, obtained by concatenating the tangential field component types respectively, directly correspond to the dimensions of the partitioned coefficient submatrices, eliminating the need for additional dimension expansion or format conversion of the components and avoiding unnecessary matrix operation overhead. During the calculation of the reflection coefficient matrix, the reflection coefficients corresponding to s-polarization and p-polarization can be directly extracted from the dual-polarization coefficient results output in a single step, without repeating the field strength derivation step, further extending the efficiency advantage of joint solution.

[0089] Specifically, in step S41, the first tangential field stitching matrix and the second tangential field stitching matrix obtained by horizontal stitching are respectively used... and Representation. The first tangential field splicing matrix. It is composed of the electric field component of the first incident layer Second incident layer electric field component The second tangential field splicing matrix is ​​formed by horizontal splicing. It is composed of the magnetic field component of the first incident layer Second incident layer magnetic field component It is made by horizontal splicing.

[0090] Specifically, in step S42, the upper and lower half-coefficient submatrices obtained by row-by-row partitioning are respectively used... and express.

[0091] Specifically, in step S43, the specific formula for calculating the first reflection coefficient matrix under the s-polarization state is as follows: ; in, Let be the first reflection coefficient matrix under the s-polarization state, specifically an L×2 dimensional vector, where L is the total diffraction order; and These are the upper and lower coefficient submatrices, respectively. The first tangential field splicing matrix is ​​specifically an L×2 dimensional vector; The specific formula for calculating the second reflection coefficient matrix under the p-polarization state is as follows: ; in, This is the second reflection coefficient matrix under p-polarization state. G is the second tangential field splicing matrix, which consists of L×2 dimensional vectors; 11 G 12 G 21 and G 22 The four global backscattering matrices obtained by partitioning the global boundary matrix of the highest physics layer are: A submatrix of dimension j; j is the imaginary unit. It is the inverse of the incident layer impedance matrix in the incident layer characteristic matrix pair.

[0092] In the above method for calculating the first and second reflection coefficient matrices, all calculation processes only need to perform basic matrix operations on the sub-matrices after partitioning, without the need for repeated calculations of the full-dimensional matrix. This further maintains the efficiency advantage of low-dimensional calculations. Finally, the reflection coefficient values ​​under the four polarization combinations of p-polarized incident → s-polarized outgoing, p-polarized incident → p-polarized outgoing, s-polarized incident → s-polarized outgoing, and s-polarized incident → p-polarized outgoing can be directly extracted from the result matrix. All polarization-related parameters are output at once, without the need for repeated calculations for different polarization combinations.

[0093] Preferably, step S5 includes: S51: Extract the zero-order component at the Nth position from the first reflection coefficient matrix and the second reflection coefficient matrix respectively, and obtain the zero-order reflection coefficient of the s component and the zero-order reflection coefficient of the p component under the p polarization state, and the zero-order reflection coefficient of the p component and the zero-order reflection coefficient of the s component under the s polarization state respectively. S52: Based on the extracted zero-order reflection coefficients of the s component and p component under the p polarization state, as well as the zero-order reflection coefficients of the p component and s component under the s polarization state, the Jones matrix of the multilayer grating structure is assembled.

[0094] The zero-order component at the Nth position in both the first and second reflection coefficient matrices represents the reflection coefficient of the outgoing polarization component of the zero-order diffracted light under different polarization incident conditions. Since all higher-order diffraction components do not participate in the Jones matrix calculation, only the zero-order component satisfies the calculation requirements of the Mueller and Jones matrices. Therefore, it is not necessary to retain the reflection coefficient results of all diffraction orders; the Jones matrix of the multilayer grating structure can be obtained directly by extracting the zero-order component at the corresponding position and assembling it. This completes the polarization optimization calculation of the semiconductor under test in the RCWA analysis process, providing a complete and efficient polarization parameter output for the optical measurement of the semiconductor under test. This output can be directly used for subsequent structural parameter inversion and optical property analysis without additional polarization component conversion or repeated calculations, avoiding the redundant overhead of storing and calculating higher-order components, and further compressing the overall calculation process from incident to result output. This extraction and assembly process has a clear logic, with all component correspondences clearly defined, preventing polarization component confusion or mismatch, and ensuring the accuracy and reliability of the Jones matrix calculation results.

[0095] Specifically, in step S51, the extracted zero-order reflection coefficients of the s component and p component under the p-polarization state, as well as the zero-order reflection coefficients of the p component and s component under the s-polarization state, are respectively used as... , , and express.

[0096] In step S52, the specific expression of the assembled Jones matrix is ​​as follows: ; in, For the Jones matrix, and These are the zero-order reflection coefficients of the p-component and s-component, respectively, under the p-polarization state. and denoted as the zero-order reflection coefficients of the p-component and s-component respectively under the s-polarization state, and N is the harmonic cutoff number.

[0097] In the Jones matrix above, This reflects the amplitude response coefficient under the polarization combination of p-polarized incident and p-polarized outgoing signals. The amplitude response coefficient corresponding to s-polarized incident → p-polarized outgoing. The amplitude response coefficient corresponding to p-polarized incident → s-polarized outgoing. The amplitude response coefficients corresponding to s-polarized incident to s-polarized outgoing polarization are assembled in one go, fully covering all polarization conversion scenarios. The result format directly meets the input requirements for subsequent optical characteristic analysis without requiring additional adjustments. Through the joint optimization of this method, the entire polarization calculation process only requires one LU decomposition and one matrix solution to obtain the complete Jones matrix. Compared with the traditional scheme of calculating polarization twice, this significantly reduces computational redundancy and compresses the overall computation time.

[0098] Preferably, after obtaining the dual-polarization forward field coefficient matrix in step S3, the method further includes: S6: Using matrix-matrix multiplication, based on the global boundary matrix of each physical layer, as well as the single-layer state matrix and single-layer propagation matrix, the dual-polarization forward field coefficient matrix is ​​recursively derived layer by layer from the incident layer to the substrate layer, thereby obtaining the dual-polarization field coefficient matrix of each physical layer.

[0099] In the RCWA analysis of the semiconductor under test, in addition to calculating the Jones matrix of the entire structure based on the reflection coefficient matrix, it is also necessary to obtain the optical field distribution information of each physical layer inside the multilayer grating structure. This internal optical field information is of great significance for analyzing characteristics such as local optical field enhancement and structural layer energy absorption. In this scheme, after obtaining the dual-polarization forward field coefficient matrix, tensor backpropagation can be achieved by only one layer-by-layer recursion in the form of matrix multiplication. This directly yields the dual-polarization field coefficient matrix of each physical layer, which simultaneously contains both p-polarization and s-polarization incident conditions. This completes the energy propagation calculation of the two polarization components in the multilayer medium, without needing to repeat the recursion process separately for each polarization. While outputting the internal optical field distribution, it still retains the efficiency advantage of joint polarization calculation. The recursion process is executed in reverse from the highest physical layer to the incident layer. Each step only uses the pre-calculated and stored global boundary matrix, single-layer state matrix, and propagation matrix of each physical layer. No additional decomposition or solution operations are required, resulting in extremely low computational overhead. Ultimately, the dual polarization field coefficient matrix output by each physical layer also contains the results of two polarization incident events, which can be directly used to extract the internal field distribution under the corresponding polarization, meeting the needs of different analysis scenarios.

[0100] Further, step S6 includes: For the highest physical layer, matrix-matrix multiplication is used to perform backward recursion on the dual-polarization forward field coefficient matrix based on the global boundary matrix, single-layer state matrix, and single-layer propagation matrix of the highest physical layer, to obtain the dual-polarization field coefficient matrix of the highest physical layer. The specific formula for calculating the dual polarization field coefficient matrix of the highest physical layer is as follows: ; in, This is the dual polarization field coefficient matrix of the highest physical layer. Let be the dual-polarization forward field coefficient matrix. These are the components in the single-layer state matrix of the highest physical layer. This is the single-layer propagation matrix of the highest physical layer. This is the recursive coefficient matrix of the highest physical layer. It is the inverse of the global forward cumulative scattering matrix in the global boundary matrix of the highest physical layer; For each current physical layer from the second highest physical layer to the lowest physical layer, matrix-matrix multiplication is used layer by layer. Based on the global boundary matrix, single-layer state matrix and single-layer propagation matrix of the previous physical layer, the dual polarization field coefficient matrix of the previous physical layer is recursively derived to obtain the dual polarization field coefficient matrix of each current physical layer. The specific formula for calculating the dual polarization field coefficient matrix of the l-th physical layer is: ; in, Let be the dual polarization field coefficient matrix of the l-th physical layer. This is the dual polarization field coefficient matrix of the (l+1)th physical layer. These are the components in the single-layer state matrix of the (l+1)th physical layer. This is the single-layer propagation matrix of the (l+1)th physical layer. This is the recursive coefficient matrix for the (l+1)th physical layer. is the inverse of the global forward cumulative scattering matrix in the global boundary matrix pair of the (l+1)th physical layer; l is an integer satisfying 1≤l≤t-1, and t is the total number of physical layers in the multilayer grating structure; For the substrate layer, matrix-matrix multiplication is used to backward recursively derive the dual polarization field coefficient matrix of the lowest physical layer based on the global boundary matrix, single-layer state matrix, and single-layer propagation matrix of the lowest physical layer, so as to obtain the dual polarization field coefficient matrix of the substrate layer. The specific formula for calculating the dual polarization field coefficient matrix of the substrate is as follows: ; in, The matrix represents the dual polarization field coefficients of the substrate. This is the dual polarization field coefficient matrix of the lowest physical layer. These are the components in the single-layer state matrix of the lowest physical layer. This is the single-layer propagation matrix of the lowest physical layer. This is the recursive coefficient matrix of the lowest physical layer. The inverse of the global forward cumulative scattering matrix in the global boundary matrix of the lowest physical layer. In the aforementioned tensor backpropagation process, since the dual-polarization forward field coefficient matrix obtained in the previous steps represents the overall dual-polarization incident coefficient result on the incident layer side, during the backpropagation, for the highest physical layer, it can directly perform a matrix multiplication with the dual-polarization forward field coefficient matrix using its pre-calculated single-layer state matrix, single-layer propagation matrix, and recursive coefficient matrix to obtain the dual-polarization field coefficient of this layer; then, for each physical layer from the second highest to the lowest physical layer, each layer only relies on the result already calculated by the previous layer, and combines its own pre-stored matrix parameters to complete a matrix multiplication calculation to obtain the field coefficient matrix of the corresponding physical layer under the two polarization incident conditions; finally, for the substrate layer, it can also use the corresponding matrix parameters pre-calculated and stored by the lowest physical layer to directly calculate the dual-polarization field coefficient matrix of the substrate layer based on the result of the lowest physical layer. The entire recursive process only requires performing matrix multiplication once layer by layer in descending order. It does not require additional time-consuming operations such as decomposition and inversion. All calculations are completed through low-complexity matrix multiplication. Moreover, a single recursion can simultaneously obtain the field coefficient results of each physical layer under both p and s polarization incident conditions. It does not require two separate polarization recursions, thus preserving the efficiency advantage of polarization joint calculation to the greatest extent. At the same time, the implementation of matrix operations is more compatible with the existing hardware parallel acceleration framework, which can further improve the overall calculation speed.

[0101] In this embodiment, the polarization optimization calculation method described herein optimizes multiple matrices in the traditional RCWA polarization calculation process, and the optimization effect is shown in Table 1 below.

[0102] Table 1. Matrix optimization effect of the method in this embodiment compared to traditional methods.

[0103] This embodiment uses a complete polarization optimization calculation method to simulate the trapezoidal multilayer grating structure. During simulation testing, the sidewall angle was set to 85 degrees, the trapezoidal height to 200 nm, and the number of structural slices to 15. The simulation results are as follows: Figure 2 As shown. In Figure 2 In the diagram, the horizontal axis represents the diffraction order used in the simulation calculation, and the vertical axis represents the speedup ratio. From Figure 2 It can be seen that the larger the diffraction order, the greater the overhead of diagonalization and inversion, and the closer the speedup ratio is to the theoretical speedup limit of 2.

[0104] Based on the above matrix optimization and simulation results, the complete polarization calculation optimization method in this embodiment has the following beneficial effects.

[0105] First, the computational complexity of the algorithm is significantly reduced. Let the total number of physical layers in the grating structure be t, and the number of truncated substrates (i.e., the total diffraction orders) be... .

[0106] The total computational cost of the traditional architecture is: ; in, The total computational cost of the traditional architecture, Let be the number of floating-point operations required to perform eigenvalue decomposition on a matrix of dimension L, Inv(2L) be the number of floating-point operations required to invert a matrix of dimension 2L, Solve(2L) be the number of floating-point operations required to solve a system of linear equations of dimension 2L, and Gemv(2L) be the number of floating-point operations required to perform matrix-vector multiplication between a matrix of dimension 2L and a vector of dimension 2L; where , .

[0107] The total computational cost of the optimized architecture provided in this embodiment is: ; in, To optimize the overall computational load of the architecture, To calculate the floating-point operations required to solve a system of linear equations with two right-hand sides for a coefficient matrix of dimension 2L, Gemm(2L,2) represents the floating-point operations required for matrix multiplication of a 2L matrix with dimensions 2L and a 2L×2 matrix. Solve_MultiColumn(2L,2) incurs only a minimal additional overhead compared to Solve(2L), and the complexity of Gemm(2L,2) is far lower than the sum of two independent Gemv(2L). Comparing the two, it can be seen that this optimized architecture merges all repetitive calculations involving polarization into a single joint calculation. Since the execution frequency of the core energy-intensive operations—eigenvalue decomposition and matrix cascading—is directly halved, nearly half of the high-complexity eigenvalue decomposition, matrix inversion, and linear equation solving operations are eliminated. When the harmonic cutoff number L increases with the demand for high-precision semiconductor measurement, the overall floating-point operation volume is compressed by nearly half, and the theoretical calculation time converges to about 50% of the original time. Theoretically, it can achieve a computational speedup of nearly 2 times, which is completely consistent with the simulation results.

[0108] Second, there is no significant increase in memory usage. This embodiment only expands the dimension of the single column vector corresponding to polarization in the original architecture to a double column matrix. The expanded dimension only changes from 2L×1 to 2L×2. The additional memory usage is only 2L floating-point numbers. Compared with the large number of high-dimensional matrices pre-stored in each physical layer of the multi-layer structure during the entire RCWA calculation process, the increase in memory is extremely small and will not significantly burden the overall memory consumption, thus adapting to the calculation needs of large-scale complex multi-layer grating structures.

[0109] Third, the accuracy of the calculation results is not lost. This embodiment optimizes the algorithm by merging repetitive calculation processes and adjusting the matrix storage structure. All calculation steps retain the calculation accuracy of the original RCWA method. No approximation or truncation operations are introduced. The final Jones matrix and internal field distribution results are completely consistent with the results obtained by traditional polarization calculation. The calculation accuracy is not sacrificed due to the increase in speed.

[0110] Fourth, hardware-level underlying storage and execution throughput optimization.

[0111] Traditional backpropagation recursion uses matrix-vector multiplication (GEMV), a BLAS Level 2 operation with a low memory access ratio (the ratio of computational cost to memory access cost), and its computational power is limited by memory bandwidth. This embodiment completely converts the recursion method to matrix-matrix multiplication (GEMM), a BLAS Level 3 operation, achieving a leap to BLAS level. Modern CPUs' high-level caches (L1 / L2 cache) can handle this leap. The matrix blocking is loaded into the register, simultaneously serving two columns of physical polarization data. This results in extremely high data reuse, successfully transforming the computational model into one where computational power is limited by the core frequency (Compute-Bound), greatly activating the SIMD / AVX512 vectorized parallel hardware throughput of modern processors.

[0112] Furthermore, when solving a system of linear equations, handling multiple right-hand side terms means that only the coefficient matrix needs to be processed. Perform a single LU matrix decomposition. Traditional methods perform two complete LU decompositions, while this embodiment reduces the overhead of the second decomposition to zero, greatly reducing the redundant computation overhead in the solution stage. The LU decomposition result can be reused for both p-polarization and s-polarization solution processes, further aligning with the underlying hardware's cache scheduling logic and improving the overall computational efficiency.

[0113] Example 2 A polarization calculation optimization system for RCWA is applied to a semiconductor under test, wherein the semiconductor under test is pre-divided into an incident layer, a substrate layer and a multilayer grating structure including multiple physical layers using the RCWA method. The system is applied in the polarization calculation optimization method for RCWA in Embodiment 1, such as... Figure 3 As shown, it includes: The parameter preprocessing module is used to obtain the optical incident parameters and grating geometric parameters of the multilayer grating structure, and to perform a single environment initialization and operator cascade based on the optical incident parameters and optical geometric parameters to obtain the system control matrix. The excitation splicing module is used to construct the incident excitation vectors under the dual polarization states respectively, and to horizontally splice the incident excitation vectors under the dual polarization states to obtain a two-column joint excitation matrix; The parallel solution module is used to perform a parallel linear solution on the two right-hand terms of the two-column joint excitation matrix using the system control matrix to obtain the dual polarization forward field coefficient matrix. The matrix calculation module is used to obtain the reflection coefficient matrix under the dual polarization state by matrix-matrix multiplication in parallel calculation based on the dual polarization forward field coefficient matrix; and to assemble the Jones matrix of the multilayer grating structure according to the reflection coefficient matrix under the dual polarization state.

[0114] In this embodiment, by obtaining the optical incident parameters and grating geometric parameters of the multilayer grating structure based on the RCWA method, complete known conditions can be provided for subsequent calculations and analyses. Through the parameter preprocessing module, a single environment initialization and operator cascading are performed based on the optical incident parameters and grating geometric parameters, enabling the one-time construction of the system control matrix. This avoids the redundant operations of repeatedly performing environment initialization and operator cascading in traditional polarization-separation calculations, reducing repetitive computations from the computational source. Simultaneously, the constructed system control matrix is ​​polarization-independent, satisfying the requirement of one decomposition and multiple solutions. Based on this system control matrix, subsequent parallel solution modules can be used to obtain the dual-polarization forward field coefficient matrix simply by performing parallel linear solutions on the two right-hand sides of the dual-column joint excitation matrix, without needing to perform parallel linear solutions on the two polarizations. The system control matrix construction and decomposition operations are performed separately for each oscillating state, significantly reducing computational redundancy. Furthermore, when constructing the dual-column joint excitation matrix, only the incident excitation vectors under dual polarization states need to be horizontally spliced ​​through the excitation splicing module to achieve parallel computation of dual polarization, without the need for additional complex dimension expansion operations, which greatly improves computational efficiency. After calculating the dual polarization forward field coefficient matrix, the reflection coefficient matrix under dual polarization states is obtained through matrix-matrix parallel computation based on the dual polarization forward field coefficient matrix, and then assembled to obtain the Jones matrix used to describe the polarization characteristics of the multilayer grating. This transforms the original two independent vector-matrix operations into a single matrix-matrix operation, which can fully utilize the parallel computing capabilities of modern computing hardware and further compress the overall computation time.

[0115] The polarization calculation optimization system for RCWA in this embodiment constructs a polarization-independent system control matrix in one go, combined with a design for parallel solution of dual polarization excitations. This eliminates the redundant steps of repeatedly constructing and decomposing the system matrix during polarization-specific calculations, and fully leverages the parallel computing performance of modern hardware. While ensuring the physical robustness of the algorithm and the consistency of calculation accuracy, it greatly improves the overall efficiency of polarization calculations. At the same time, it achieves a balance between memory usage and computing power consumption, is adaptable to computing devices with different hardware configurations, and can meet the calculation needs of polarization-related optical measurements in the semiconductor field.

[0116] The functions of each module in the polarization calculation optimization system for RCWA described in this embodiment are the same as the method steps of the polarization calculation optimization method for RCWA described in Embodiment 1. Therefore, for details not covered in this embodiment, please refer to Embodiment 1 and... Figures 1 to 2 The specific details will not be repeated here.

[0117] Example 3 This embodiment also provides a polarization calculation optimization device for RCWA, including a processor, a memory, and a computer program stored in the memory and run on the processor. When the computer program runs, it implements the method steps in the polarization calculation optimization method for RCWA of Embodiment 1.

[0118] By using a computer program stored in memory and running on a processor, the overall efficiency of polarization calculation is greatly improved while ensuring the physical robustness of the algorithm and the consistency of calculation accuracy. At the same time, it achieves a balance between memory usage and computing power consumption, can be adapted to computing devices with different hardware configurations, and can meet the calculation needs of polarization-related optical measurements in the semiconductor field.

[0119] The processor 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, discrete gate or transistor logic devices, discrete hardware components, etc. A general-purpose processor can be a microprocessor or any conventional processor. The processor is the control center of the computer device, connecting all parts of the computer device through various interfaces and lines.

[0120] Memory can be used to store computer programs and / or models. The processor performs various functions of the computer device by running or executing the computer programs and / or models stored in the memory, and by accessing data stored in the memory. Memory can primarily include a program storage area and a data storage area. The program storage area can store the operating system and at least one application program required for a function (e.g., sound playback, image playback, etc.); the data storage area can store data created based on the use of the mobile phone (e.g., audio data, video data, etc.). Furthermore, memory can include high-speed random access memory, and can also include non-volatile memory, such as hard disks, RAM, plug-in hard disks, SmartMedia Cards (SMC), Secure Digital (SD) cards, Flash Cards, at least one disk storage device, flash memory device, or other volatile solid-state storage devices.

[0121] It should be understood that each block of a flowchart and / or block diagram, and combinations of blocks in a flowchart and / or block diagram, can be implemented by a computer program. These computer programs can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing device to produce a machine, such that instructions executable by the processor of the computer or other programmable data processing device generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0122] These computer programs may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0123] These computer programs may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0124] This embodiment also provides a computer storage medium, which includes at least one instruction that, when executed by a computer, implements the method steps in the polarization calculation optimization method for RCWA of Embodiment 1.

[0125] By executing a computer storage medium containing at least one instruction, the overall efficiency of polarization calculation is greatly improved while ensuring the physical robustness of the algorithm and the consistency of calculation accuracy. At the same time, it achieves a balance between memory usage and computing power consumption, can be adapted to computing devices with different hardware configurations, and can meet the calculation needs of polarization-related optical measurements in the semiconductor field.

[0126] Similarly, for details not covered in this embodiment, please refer to Embodiment 1, Embodiment 2, and... Figures 1 to 3 The specific details will not be repeated here.

[0127] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A polarization calculation optimization method for RCWA, characterized in that, It is applied to the semiconductor under test, and the semiconductor under test is pre-divided into an incident layer, a substrate layer and a multilayer grating structure including multiple physical layers using the RCWA method; The method includes: The optical incident parameters and grating geometric parameters of the multilayer grating structure are obtained, and a single environment initialization and operator cascade are performed based on the optical incident parameters and optical geometric parameters to obtain the system control matrix; Construct the incident excitation vectors under the dual polarization states respectively, and then horizontally concatenate the incident excitation vectors under the dual polarization states to obtain the two-column joint excitation matrix; Using the system control matrix, the two right-hand terms of the two columns of the bi-column joint excitation matrix are solved linearly in parallel to obtain the bipolar forward field coefficient matrix; Based on the aforementioned dual-polarization forward field coefficient matrix, the reflection coefficient matrix under the dual-polarization state is obtained by parallel matrix-matrix multiplication. The Jones matrix of the multilayer grating structure is obtained by assembling the reflection coefficient matrix under dual polarization states.

2. The polarization calculation optimization method for RCWA according to claim 1, characterized in that, The physical layers of the multilayer grating structure increase sequentially from low to high; Based on the optical incident parameters and optical geometric parameters, a single environmental initialization and operator cascade are performed to obtain the system control matrix, including: The harmonic cutoff number is preset, and the total diffraction order and diffraction order index are determined according to the harmonic cutoff number; Based on the total diffraction order and the diffraction order index, the substrate is constructed according to the optical incident parameters, and the normalized wave vector diagonal matrix pair, azimuth diagonal matrix pair, incident layer feature matrix pair and substrate layer feature matrix pair are calculated respectively. Traverse all physical layers of the multilayer grating structure, perform eigenlayer loops based on the grating geometric parameters, normalized wave vector diagonal matrix pairs, and azimuth diagonal matrix pairs, and construct the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer layer by layer. The substrate is initialized based on the substrate feature matrix pair to obtain the corresponding substrate boundary matrix pair. Then, based on the substrate boundary matrix pair and the single-layer state matrix and single-layer propagation matrix of each physical layer, the global boundary matrix pair corresponding to each physical layer is obtained by recursively deriving from the substrate to the incident layer. The system control matrix is ​​obtained by assembling the global boundary matrix pair corresponding to the highest physical layer and the incident layer feature matrix pair.

3. The polarization calculation optimization method for RCWA according to claim 2, characterized in that, The optical incident parameters include incident wavelength, incident polar angle, azimuth angle, incident layer refractive index, and substrate layer refractive index; The grating geometry parameters include the thickness and dielectric constant distribution of each physical layer; Traversing all physical layers of the multilayer grating structure, and performing eigenlayer iterations based on the grating geometric parameters, normalized wave vector diagonal matrix pairs, and azimuth diagonal matrix pairs, layer by layer, the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer are constructed, including: Select the l-th physical layer and construct the corresponding Toplitz matrix pair based on the dielectric constant distribution of the selected physical layer; Based on the normalized wave vector diagonal matrix pair and the constructed Topplitz matrix pair, operator matrices are constructed for the TE polarization mode and the TM polarization mode respectively, and the operator matrix pair corresponding to the selected physical layer is obtained. The selected physical layer operator matrix pairs are decomposed into eigenvalues ​​to obtain the corresponding combination of eigenma matrices. Based on the combination of the azimuth diagonal matrix pair and the characteristic matrix of the selected physical layer, the field component coupling matrix is ​​constructed; and the constructed field component coupling matrix is ​​assembled into the single-layer state matrix of the corresponding physical layer. Based on the selected physical layer thickness and the corresponding feature matrix combination, a propagation component matrix is ​​constructed; and the constructed propagation component matrix is ​​assembled into a single-layer propagation matrix of the corresponding physical layer. By traversing each physical layer of the multilayer grating structure, the single-layer state matrix and single-layer propagation matrix corresponding to each physical layer are obtained layer by layer in the same way.

4. The polarization calculation optimization method for RCWA according to claim 3, characterized in that, The normalized wave vector diagonal matrix pair includes the x-direction wave vector diagonal matrix and the y-direction wave vector diagonal matrix; Each physical layer’s Toplitz matrix pair includes a dielectric constant matrix and an inverse dielectric constant matrix, and each physical layer’s eigenma matrix combination includes a first eigenvector matrix, a second eigenvector matrix, a first eigenvalue diagonal matrix, and a second eigenvalue diagonal matrix. For the l-th physical layer, the specific calculation formula for the operator matrix pair is as follows: ； ； in, and These are all components of the operator matrix pair of the l-th physical layer, specifically the TE mode operator matrix and the TM mode operator matrix, respectively; P is the inverse permittivity matrix of the l-th physical layer. y is the wave vector diagonal matrix; A and B are components of the intermediate matrix pair constructed based on the wave vector diagonal matrix in the x direction and the dielectric constant matrix of the l-th physical layer, specifically the first intermediate matrix and the second intermediate matrix, respectively. The specific calculation formulas for the first intermediate matrix A and the second intermediate matrix B of the l-th physical layer are as follows: ； ； in, Let x be the diagonal matrix of the wave vector in the x-direction. Let l be the dielectric constant matrix of the l-th physical layer. It is the inverse matrix of the dielectric constant matrix of the l-th physical layer; For the l-th physical layer, the specific formula for calculating the feature matrix combination is as follows: ； ； Where M1 and M2 are the first eigenvector matrix and the second eigenvector matrix in the feature matrix combination of the l-th physical layer, respectively, and Q1 and Q2 are the first eigenvalue diagonal matrix and the second eigenvalue diagonal matrix in the feature matrix combination of the l-th physical layer, respectively. For the l-th physical layer, the specific formula for calculating the field component coupling matrix is ​​as follows: ； ； ； ； ； ； ； ； Among them, V ss V sp V ps V pp W ss W sp W ps and W pp All are field components in the field component coupling matrix of the l-th physical layer; F c and F s These are all components of the diagonal matrix pair of the azimuth angles of the l-th physical layer, specifically the cosine direction matrix and the sine direction matrix, respectively; V 11 V 12 V 21 and V 22 All are coupling submatrices of the l-th physical layer; The specific formula for calculating the coupling submatrix of the l-th physical layer is: ； ； ； ； Among them, A -1 and B -1 These are the inverse matrices of the first intermediate matrix A and the second intermediate matrix B of the l-th physical layer, respectively. Each encapsulated physical layer's single-layer state matrix includes a mode coupling matrix and a field coupling matrix. The specific expressions for the mode coupling matrix and field coupling matrix of the l-th physical layer are as follows: ； ； Among them, V l and W l These are the mode coupling matrix and field coupling matrix of the l-th physical layer, respectively; The specific formula for calculating the propagation component matrix of the l-th physical layer is: ； ； Where X1 and X2 are both propagation component matrices of the l-th physical layer, d l Let be the thickness of the l-th physical layer, and k0 be the preset free space wave vector; The specific expression for the single-layer propagation matrix of the l-th physical layer, which is encapsulated, is as follows: ； Among them, X l Let l be the single-layer propagation matrix of the l-th physical layer. Let L be an L×L zero matrix, where L is the total diffraction order.

5. The polarization calculation optimization method for RCWA according to claim 2, characterized in that, The substrate boundary matrix includes the substrate forward scattering matrix and the substrate back scattering matrix; The specific expression for the substrate boundary matrix pair is: ； ； in, and Y1 and Z2 are the substrate forward scattering and substrate back scattering matrices, respectively, with j being the imaginary unit; Y2 and Z2 are components of the substrate characteristic matrix pair, specifically the substrate admittance matrix and substrate impedance matrix, respectively; I L It is an L×L dimensional identity matrix. Let L be an L×L zero matrix, where L is the total diffraction order; Each physical layer's global boundary matrix pair includes a global forward cumulative scattering matrix and a global backward cumulative scattering matrix; The specific expression for the global boundary matrix of the l-th physical layer is: ； ； in, and V represents the global forward cumulative scattering matrix and the global backward cumulative scattering matrix of the l-th physical layer, respectively. l and W l Let X be the mode coupling matrix and field coupling matrix in the single-layer state matrix of the l-th physical layer, respectively. l Let l be the single-layer propagation matrix of the l-th physical layer. Let I be the recursive coefficient matrix of the l-th physical layer; 2L for An identity matrix of 3D; The specific expression of the assembled system control matrix is ​​as follows: ； in, The system control matrix; Y1 and Z1 are components of the incident layer characteristic matrix, specifically the incident layer admittance matrix and the incident layer impedance matrix, respectively; F 11 F 12 F 21 and F 22 The global forward cumulative scattering matrix of the l-th physical layer The four obtained after partitioning A submatrix of dimension G 11 G 12 G 21 and G 22 The global backscattering matrix of the l-th physical layer The four obtained after partitioning A submatrix of dimension.

6. The polarization calculation optimization method for RCWA according to claim 2, characterized in that, The incident excitation vectors under the dual polarization states are constructed separately, and then horizontally concatenated to obtain a two-column joint excitation matrix, including: Define a zero-order diffraction selection vector, and based on the zero-order diffraction selection vector, define the first component field vector in the p-polarization state and the second component field vector in the s-polarization state respectively; Based on the first component field vector, a first single-column excitation vector in the p-polarization state is constructed; and based on the second component field vector, a second single-column excitation vector in the s-polarization state is constructed. The first single-column excitation vector and the second single-column excitation vector are horizontally concatenated along the column direction to obtain the dual-column joint excitation matrix.

7. The polarization calculation optimization method for RCWA according to claim 6, characterized in that, Both the first and second component field vectors include four tangential field components. The specific expressions for the four tangential field components in the first component field vector are as follows: ； in, , , and These are the four tangential field components in the first component field vector. Choosing a vector for zero-order diffraction and These are the polar angle of incidence and the refractive index of the incident layer, respectively, in the optical incident parameters; The specific expressions for the four tangential field components in the second component field vector are as follows: ； in, , , and These are the four tangential field components in the second component field vector; The specific expressions for the constructed first and second single-column activation vectors are as follows: ； ； in, and These are the first and second single-column excitation vectors, respectively, both of which are L×1 dimensional column vectors, where L is the total diffraction order; and All of these are components of the incident layer characteristic matrix calculated based on the optical incident parameters, specifically the incident layer admittance matrix and the incident layer impedance matrix; j is the imaginary unit. The specific expression for the concatenated bi-column joint excitation matrix is ​​as follows: ； in, It is a two-column joint activation matrix, specifically a 2L×2 dimensional column vector.

8. The polarization calculation optimization method for RCWA according to claim 7, characterized in that, The first component field vector under p-polarization state specifically includes the electric field component of the first incident layer. and the first incident layer magnetic field component The second component field vector in the s-polarization state specifically includes the second incident layer electric field component. Second incident layer magnetic field component ; Based on the aforementioned dual-polarization forward field coefficient matrix, the reflection coefficient matrix under dual polarization states is obtained through parallel matrix-matrix multiplication calculations, including: According to the column direction, the first incident layer electric field component in the first component field vector and the second incident layer electric field component in the second component field vector are horizontally spliced ​​together to obtain the first tangential field splicing matrix; and the first incident layer magnetic field component in the first component field vector and the second incident layer magnetic field component in the second component field vector are horizontally spliced ​​together to obtain the second tangential field splicing matrix. Divide the dual-polarization forward field coefficient matrix into rows to obtain the upper half coefficient submatrix and the lower half coefficient submatrix; The first reflection coefficient matrix in the s-polarization state is calculated based on the first tangential field splicing matrix, the upper half-coefficient sub-matrix, the lower half-coefficient sub-matrix, and the global boundary matrix pair of the highest physical layer; and the second reflection coefficient matrix in the p-polarization state is calculated based on the second tangential field splicing matrix, the upper half-coefficient sub-matrix, the lower half-coefficient sub-matrix, the global boundary matrix pair of the highest physical layer, and the incident layer feature matrix pair.

9. The polarization calculation optimization method for RCWA according to claim 8, characterized in that, The specific formula for calculating the first reflection coefficient matrix under s-polarization is: ； in, Let be the first reflection coefficient matrix under the s-polarization state, specifically an L×2 dimensional vector, where L is the total diffraction order; and These are the upper and lower coefficient submatrices, respectively. The first tangential field splicing matrix is ​​specifically an L×2 dimensional vector; The specific formula for calculating the second reflection coefficient matrix under the p-polarization state is as follows: ； in, This is the second reflection coefficient matrix under p-polarization state. G is the second tangential field splicing matrix, which consists of L×2 dimensional vectors; 11 G 12 G 21 and G 22 The four global backscattering matrices obtained by partitioning the global boundary matrix of the highest physics layer are: A submatrix of dimension j; j is the imaginary unit. It is the inverse of the incident layer impedance matrix in the incident layer characteristic matrix pair.

10. The polarization calculation optimization method for RCWA according to claim 9, characterized in that, Based on the reflection coefficient matrix under dual polarization states, the Jones matrix of the multilayer grating structure is assembled, including: The zero-order component at the Nth position is extracted from the first reflection coefficient matrix and the second reflection coefficient matrix respectively, and the zero-order reflection coefficient of the s component and the zero-order reflection coefficient of the p component under the p polarization state, and the zero-order reflection coefficient of the p component and the zero-order reflection coefficient of the s component under the s polarization state are obtained respectively. Based on the extracted zero-order reflection coefficients of the s component and p component under the p polarization state, as well as the zero-order reflection coefficients of the p component and s component under the s polarization state, the Jones matrix of the multilayer grating structure is assembled. The specific expression of the assembled Jones matrix is ​​as follows: ； in, For the Jones matrix, and These are the zero-order reflection coefficients of the p-component and s-component, respectively, under the p-polarization state. and denoted as the zero-order reflection coefficients of the p-component and s-component respectively under the s-polarization state, and N is the harmonic cutoff number.

11. The polarization calculation optimization method for RCWA according to claim 2, characterized in that, Using the system control matrix, the two right-hand side terms of the two-column joint excitation matrix are solved linearly in parallel to obtain the dual-polarization forward field coefficient matrix, including: The linear equation solver is invoked to perform a single LU decomposition on the system control matrix to obtain the LU decomposition result. Based on the LU decomposition results, forward and backward permutations are performed in parallel on the two right-hand terms of the bi-column joint excitation matrix to obtain the bipolar forward field coefficient matrix.

12. The polarization calculation optimization method for RCWA according to claim 11, characterized in that, The specific expression for the matrix equation used to solve the dual-polarization forward field coefficient matrix is ​​as follows: ； in, The dual-polarization forward field coefficient matrix is ​​specifically a 2L×2 dimensional column vector, where L is the total diffraction order; FG is the system control matrix, and D... mat Let be the bi-column joint excitation matrix.

13. The polarization calculation optimization method for RCWA according to claim 11, characterized in that, After obtaining the dual-polarization forward field coefficient matrix, the method further includes: Using matrix-matrix multiplication, based on the global boundary matrix of each physical layer, as well as the single-layer state matrix and single-layer propagation matrix, the dual-polarization forward field coefficient matrix is ​​recursively derived layer by layer from the incident layer to the substrate layer, thereby obtaining the dual-polarization field coefficient matrix of each physical layer.

14. The polarization calculation optimization method for RCWA according to claim 13, characterized in that, Based on the global boundary matrix of each physical layer, as well as the single-layer state matrix and single-layer propagation matrix, the dual-polarization forward field coefficient matrix is ​​recursively derived layer by layer backward from the incident layer to the substrate layer, including: For the highest physical layer, matrix-matrix multiplication is used to perform backward recursion on the dual-polarization forward field coefficient matrix based on the global boundary matrix, single-layer state matrix, and single-layer propagation matrix of the highest physical layer, to obtain the dual-polarization field coefficient matrix of the highest physical layer. The specific formula for calculating the dual polarization field coefficient matrix of the highest physical layer is as follows: ； in, This is the dual polarization field coefficient matrix of the highest physical layer. Let be the dual-polarization forward field coefficient matrix. These are the components in the single-layer state matrix of the highest physical layer. This is the single-layer propagation matrix of the highest physical layer. This is the recursive coefficient matrix of the highest physical layer. It is the inverse of the global forward cumulative scattering matrix in the global boundary matrix of the highest physical layer; For each current physical layer from the second highest physical layer to the lowest physical layer, matrix-matrix multiplication is used layer by layer. Based on the global boundary matrix, single-layer state matrix and single-layer propagation matrix of the previous physical layer, the dual polarization field coefficient matrix of the previous physical layer is recursively derived to obtain the dual polarization field coefficient matrix of each current physical layer. The specific formula for calculating the dual polarization field coefficient matrix of the l-th physical layer is: ； in, Let be the dual polarization field coefficient matrix of the l-th physical layer. This is the dual polarization field coefficient matrix of the (l+1)th physical layer. These are the components in the single-layer state matrix of the (l+1)th physical layer. This is the single-layer propagation matrix of the (l+1)th physical layer. This is the recursive coefficient matrix for the (l+1)th physical layer. is the inverse of the global forward cumulative scattering matrix in the global boundary matrix pair of the (l+1)th physical layer; l is an integer satisfying 1≤l≤t-1, and t is the total number of physical layers in the multilayer grating structure; For the substrate layer, matrix-matrix multiplication is used to backward recursively derive the dual polarization field coefficient matrix of the lowest physical layer based on the global boundary matrix, single-layer state matrix, and single-layer propagation matrix of the lowest physical layer, so as to obtain the dual polarization field coefficient matrix of the substrate layer. The specific formula for calculating the dual polarization field coefficient matrix of the substrate is as follows: ； in, The matrix represents the dual polarization field coefficients of the substrate. This is the dual polarization field coefficient matrix of the lowest physical layer. These are the components in the single-layer state matrix of the lowest physical layer. This is the single-layer propagation matrix of the lowest physical layer. This is the recursive coefficient matrix of the lowest physical layer. It is the inverse of the global forward cumulative scattering matrix in the global boundary matrix of the lowest physical layer.

15. A polarization calculation optimization system for RCWA, characterized in that, It is applied to the semiconductor under test, and the semiconductor under test is pre-divided into an incident layer, a substrate layer and a multilayer grating structure including multiple physical layers using the RCWA method; The system is applied in the polarization calculation optimization method for RCWA as described in any one of claims 1 to 14, comprising: The parameter preprocessing module is used to obtain the optical incident parameters and grating geometric parameters of the multilayer grating structure, and to perform a single environment initialization and operator cascade based on the optical incident parameters and optical geometric parameters to obtain the system control matrix. The excitation splicing module is used to construct the incident excitation vectors under the dual polarization states respectively, and to horizontally splice the incident excitation vectors under the dual polarization states to obtain a two-column joint excitation matrix; The parallel solution module is used to perform a parallel linear solution on the two right-hand terms of the two-column joint excitation matrix using the system control matrix to obtain the dual polarization forward field coefficient matrix. The matrix calculation module is used to obtain the reflection coefficient matrix under the dual polarization state by matrix-matrix multiplication in parallel calculation based on the dual polarization forward field coefficient matrix; and to assemble the Jones matrix of the multilayer grating structure according to the reflection coefficient matrix under the dual polarization state.

16. A polarization calculation optimization device for RCWA, characterized in that, It includes a processor, a memory, and a computer program stored in the memory and executable on the processor, wherein the computer program, when executed, implements the method steps in the polarization calculation optimization method for RCWA as described in any one of claims 1 to 14.

17. A computer storage medium, characterized in that, The computer storage medium includes at least one instruction that, when executed by a computer, implements the method steps in the polarization calculation optimization method for RCWA as described in any one of claims 1 to 14.

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