Theoretical spectral data calculation method

Abstract

The invention provides a theoretical spectral data calculation method which comprises the following steps: in a theoretical spectral data calculation process, aiming at a matrix index in a coupled wave equation set in an index form, firstly converting the matrix index into b power of a product of an index of a second matrix and an index of a third matrix, respectively converting the index of the second matrix and the index of the third matrix into block matrixes, and implementing matrix diagonalization by using the matrixes in the blocks so as to convert the matrix indexes into corresponding calculation formulas; compared with the prior art in which the first matrix in the matrix indexes is directly subjected to matrix diagonalization and then the diagonalization result is substituted into the matrix indexes to obtain the calculation formula corresponding to the matrix indexes, the theoretical spectral data calculation method provided by the invention has the advantages that the dimension of the matrix needing to be subjected to matrix diagonalization is reduced; therefore, the diagonalization speed of the matrix is improved, a large amount of calculation amount is saved, and the calculation efficiency of theoretical spectral data is improved.

Description

Technical Field

[0001] The present disclosure relates to the technical field of optical critical dimension measurement, and in particular to a method for calculating theoretical spectral data. Background Art

[0002] Optical Critical Dimension (OCD) technology estimates the specific morphological parameters of a sample by acquiring the scattered signals of the periodic structure in a specific measured area of ​​the sample and the model of the sample. It can meet the needs of fast and accurate measurement in new processes and new technologies, and is non-contact and non-destructive. It is widely used in the semiconductor manufacturing industry, where the morphological parameters are also called structural parameters. The principle of optical critical dimension measurement technology can be generally described as: collecting the measured spectral data of the periodic structure on the surface of the sample, calculating the theoretical spectral data (i.e., theoretical spectrum) of the periodic structure, and by comparing the measured spectral data with the theoretical spectral data, the morphological parameters to be measured of the periodic structure, such as the critical dimension or side wall angle of the periodic structure, are obtained when the evaluation function meets the preset conditions.

[0003] In the above optical critical dimension measurement process, the algorithm used to calculate the theoretical spectral data is the Rigorous Coupled Wave Analysis (RCWA) algorithm, hereinafter referred to as the RCWA algorithm. The process of the RCWA algorithm is to substitute the dielectric constant of the sample and the Fourier series expansion of the electromagnetic field of the periodic structure into the Maxwell equations to obtain the coupled wave equations in differential form, and then calculate the theoretical spectral data based on the coupled wave equations.

[0004] However, in the existing calculation method, the amount of calculation required to solve the coupled wave equations in differential form to obtain the reflection coefficient is large, resulting in low calculation efficiency of the theoretical spectral data. Therefore, it is necessary to further optimize the calculation method of the theoretical spectral data to improve the calculation efficiency of the theoretical spectral data.

[0005] It should be noted that the information disclosed in the above background technology section is only used to enhance the understanding of the background of the present disclosure, and therefore may include information that does not constitute the prior art known to ordinary technicians in the field. Summary of the invention

[0006] In view of the problems in the prior art, the present invention aims to provide a method for calculating theoretical spectral data, which can reduce the amount of calculation of theoretical spectral data and improve the calculation efficiency of theoretical spectral data.

[0007] Specifically, the first aspect of the present disclosure provides a method for calculating theoretical spectral data, which may include the following steps:

[0008] Obtaining a coupled wave equation set in differential form for a sample having a periodic structure in both a first direction and a second direction according to a rigorous coupled wave analysis algorithm;

[0009] Integrating the coupled wave equations in the height direction of the periodic structure to obtain an exponential coupled wave equation, wherein the height direction, the first direction and the second direction are perpendicular to each other, and the exponential coupled wave equation includes a matrix index;

[0010] Get the matrix in the matrix index as the first matrix;

[0011] Converting the matrix index into the product of the index of the second matrix and the index of the third matrix raised to the power of b, so that the denominators of the fractions included in the matrices in the index of the second matrix and the index of the third matrix both include b, where b is a preset integer and b≥2;

[0012] The exponent of the second matrix is ​​converted into the first block matrix, and the exponent of the third matrix is ​​converted into the second block matrix. Based on the result of matrix diagonalization of the matrices common to the blocks of the first block matrix and the result of matrix diagonalization of the matrices common to the blocks of the second block matrix, the calculation formula corresponding to the matrix exponent is obtained, and the dimension of the matrix in each block is half of the dimension of the first matrix;

[0013] The reflection coefficient corresponding to the periodic structure is obtained according to a calculation formula (i.e., a calculation formula corresponding to the matrix index) and an electromagnetic field continuity condition on the surface of the periodic structure;

[0014] The theoretical spectrum data corresponding to the periodic structure is obtained according to the reflection coefficient.

[0015] In a possible implementation of the first aspect, integrating the coupled wave equations in the height direction of the periodic structure to obtain the coupled wave equations in exponential form comprises the following steps:

[0016] In the height direction of the periodic structure, the coupled wave equations in differential form are integrated from the lower surface height of the periodic structure to the upper surface height of the periodic structure to obtain the coupled wave equations in exponential form.

[0017] The above technical solution provides a specific implementation method for implementing integration of a periodic structure in the height direction, which can achieve the acquisition of a coupled wave equation group in exponential form.

[0018] In a possible implementation of the first aspect, the exponent of the second matrix is ​​converted into the first block matrix, the exponent of the third matrix is ​​converted into the second block matrix, and the calculation formula corresponding to the matrix exponent is obtained based on the result of matrix diagonalization of the matrix common to each block of the first block matrix and the result of matrix diagonalization of the matrix common to each block of the second block matrix, wherein the dimension of the matrix in each block is half of the dimension of the first matrix, including the following steps:

[0019] The exponent of the second matrix is ​​converted into a first block matrix, so that each block in the first block matrix is ​​a hyperbolic function of the fourth matrix, and the exponent of the third matrix is ​​converted into a second block matrix, so that each block in the second block matrix is ​​a trigonometric function of the fifth matrix. Based on the results of matrix diagonalization performed on the fourth matrix and the fifth matrix respectively, the calculation formula corresponding to the matrix exponent is obtained, wherein the dimensions of the fourth matrix and the fifth matrix are both half of the dimensions of the first matrix.

[0020] In the above technical solution, since the dimensions of the fourth matrix and the fifth matrix are both half of the dimension of the first matrix, the speed of matrix diagonalization can be improved, thereby improving the computational efficiency of theoretical spectral data calculation.

[0021] In a possible implementation of the first aspect, obtaining a coupled wave equation group in a differential form of a sample having a periodic structure in both a first direction and a second direction according to a rigorous coupled wave analysis algorithm comprises the following steps:

[0022] The magnetic field, electric field and dielectric constant corresponding to the periodic structure are expanded by Fourier series respectively;

[0023] Substitute the obtained Fourier series expansions of the magnetic field, electric field and dielectric constant into the Maxwell equations to obtain the coupled wave equations in differential form;

[0024] The mathematical expression of the coupled wave equations in differential form is as follows:

[0025]

[0026] Among them, k0 is the wave vector in vacuum environment, P and Q are the optical property matrices corresponding to the sample, and S x (z) represents the Fourier coefficient matrix corresponding to the component of the electric field in the first direction at position z, S y (z) represents the Fourier coefficient matrix corresponding to the component of the electric field in the second direction at position z, U x (z) represents the Fourier coefficient matrix corresponding to the component of the magnetic field in the first direction at position z, U y (z) represents the Fourier coefficient matrix corresponding to the component of the magnetic field in the second direction at position z, and z represents the position of the periodic structure in the height direction.

[0027] In a possible implementation of the first aspect, the mathematical expression of the coupled wave equations in exponential form is as follows:

[0028]

[0029] Wherein, 0≤z≤H, H is the distance from the lower surface of the periodic structure to the upper surface of the periodic structure in the height direction;

[0030] The mathematical expression of the first matrix is ​​as follows:

[0031]

[0032] Among them, the dimension of the first matrix is ​​4(2M+1)(2N+1), M is the truncation order of the Fourier series expansion in the first direction, N is the truncation order of the Fourier series expansion in the second direction, and exp is the natural base of the matrix exponential.

[0033] In a possible implementation of the first aspect, converting the matrix index into the b-th power of the product of the exponent of the second matrix and the exponent of the third matrix, so that the denominators of the fractions included in the matrices in the exponent of the second matrix and the exponent of the third matrix both include b, comprises the following steps:

[0034] The first matrix is ​​converted into the sum of a symmetric matrix and an antisymmetric matrix, where the mathematical expression of the symmetric matrix is ​​as follows:

[0035]

[0036] The mathematical expression of an antisymmetric matrix is ​​as follows:

[0037]

[0038] Obtain a second matrix and a third matrix according to the symmetric matrix and the antisymmetric matrix, wherein the second matrix is ​​the quotient of the symmetric matrix and b, and the third matrix is ​​the quotient of the antisymmetric matrix and b;

[0039] Among them, the mathematical expression of the second matrix is:

[0040]

[0041] The mathematical expression of the third matrix is:

[0042]

[0043] The b-th power of the product is obtained based on the exponent of the second matrix and the exponent of the third matrix. The corresponding mathematical expression is:

[0044]

[0045] In a possible implementation of the first aspect, a mathematical expression for converting the exponent of the second matrix into the first block matrix is ​​as follows:

[0046]

[0047] in, is the fourth matrix, is the hyperbolic sine function corresponding to the fourth matrix, is the hyperbolic cosine function corresponding to the fourth matrix;

[0048] The mathematical expression for converting the exponential of the third matrix into the second block matrix is ​​as follows:

[0049]

[0050] in, is the fifth matrix, is the cosine function corresponding to the fifth matrix, is the sine function corresponding to the fifth matrix.

[0051] In a possible implementation of the first aspect, dimensions of the fourth matrix and the fifth matrix are both 2(2M+1)(2N+1).

[0052] In a possible implementation of the first aspect, obtaining a calculation formula corresponding to a matrix index based on a result of performing matrix diagonalization on the fourth matrix and the fifth matrix respectively includes the following steps:

[0053] The fourth matrix and the fifth matrix are diagonalized respectively, and the mathematical expressions corresponding to the diagonalization of the fourth matrix and the fifth matrix are as follows:

[0054]

[0055] Among them, D1 and D2 are diagonal matrices, U1 and U2 are orthogonal matrices, and I is the unit matrix;

[0056] The first calculation formula corresponding to the exponent of the second matrix is ​​obtained based on the result of the diagonalization of the fourth matrix, and the second calculation formula corresponding to the exponent of the third matrix is ​​obtained based on the result of the diagonalization of the fifth matrix. The corresponding mathematical expressions are as follows:

[0057]

[0058]

[0059] Wherein, the first calculation formula is:

[0060]

[0061] The second calculation formula is:

[0062]

[0063] Convert the matrix exponent into the product of the first calculation formula and the second calculation formula to the power of b to obtain the calculation formula corresponding to the matrix exponent. The corresponding mathematical expression is as follows:

[0064]

[0065] Among them, the calculation formula corresponding to the matrix index is:

[0066]

[0067] Among them, sh(D1) is the hyperbolic sine function corresponding to the diagonal matrix D1, ch(D1) is the hyperbolic cosine function corresponding to the diagonal matrix D, cos(D2) is the cosine function corresponding to the diagonal matrix D2, and sin(D2) is the sine function corresponding to the diagonal matrix D2.

[0068] In a possible implementation of the first aspect, obtaining a reflection coefficient corresponding to the periodic structure according to a calculation formula and an electromagnetic field continuity condition on a surface of the periodic structure includes the following steps:

[0069] Substitute the calculation formula corresponding to the matrix index into the coupled wave equations in exponential form;

[0070] The reflection coefficient is obtained based on the exponential form of the coupled wave equations and the electromagnetic field continuity conditions at z=0 and z=H, where z=0 represents the upper surface position or the lower surface position of the periodic structure in the height direction; z=H represents the lower surface position or the upper surface position of the periodic structure in the height direction.

[0071] Compared with the prior art, the present invention has the following beneficial effects:

[0072] Through the technical solution provided by the present invention, in the process of calculating theoretical spectral data, for the matrix index in the coupled wave equation group in exponential form, the matrix index is first converted into the bth power of the product of the index of the second matrix and the index of the third matrix, and then the index of the second matrix and the index of the third matrix are respectively converted into block matrices, and the matrices in the blocks are used to implement matrix diagonalization, so as to convert the matrix index into the corresponding calculation formula; compared with the existing technology of directly performing matrix diagonalization on the first matrix in the matrix index and then substituting the diagonalization result into the matrix index to obtain the calculation formula corresponding to the matrix index, in the theoretical spectral data calculation method provided by the present invention, since the matrix dimension required for matrix diagonalization is reduced, the speed of matrix diagonalization is improved, a large amount of calculation is saved, and thus the calculation efficiency of theoretical spectral data is improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Other features, objects and advantages of the present disclosure will become more apparent from a reading of the detailed description of non-limiting embodiments made with reference to the following accompanying drawings.

[0074] Figure 1 A schematic flow chart of a method for calculating theoretical spectral data is provided according to an embodiment of the present disclosure.

[0075] Figure 2 According to an embodiment of the present disclosure, a flowchart for obtaining a coupled wave equation group in differential form is provided.

[0076] Figure 3 According to an embodiment of the present disclosure, a schematic cross-sectional view in the height direction of a sample having a periodic structure in both a first direction X and a second direction Y is provided.

[0077] Figure 4 According to an embodiment of the present disclosure, a flow chart of converting a matrix index into the b-th power of the product of an index of a second matrix and an index of a third matrix is ​​provided.

[0078] Figure 5 According to an embodiment of the present disclosure, a flowchart of a calculation formula for obtaining a matrix index is provided.

[0079] Figure 6 According to an embodiment of the present disclosure, a schematic flow chart of obtaining a reflection coefficient corresponding to a periodic structure is provided. DETAILED DESCRIPTION

[0080] The following describes the embodiments of the present disclosure through specific examples, and those skilled in the art can easily understand other advantages and effects of the present disclosure from the contents disclosed in the present disclosure. The present disclosure can also be implemented or applied through other different specific embodiments, and the details in the present disclosure can also be modified or changed in various ways according to different viewpoints and application systems without departing from the spirit of the present disclosure. It should be noted that the embodiments in the present disclosure and the features in the embodiments can be combined with each other without conflict.

[0081] The following is a detailed description of the embodiments of the present disclosure with reference to the accompanying drawings so that those skilled in the art can easily implement the present disclosure. The present disclosure can be embodied in many different forms and is not limited to the embodiments described herein.

[0082] In the representations of the present disclosure, the reference terms "one embodiment", "some embodiments", "examples", "specific examples", or "some examples" and the like mean that the specific features, structures, materials, or characteristics represented in conjunction with the embodiment or example are included in at least one embodiment or example of the present disclosure. Moreover, the specific features, structures, materials, or characteristics represented may be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art may combine and combine different embodiments or examples represented in the present disclosure and the features of different embodiments or examples, unless they are mutually contradictory.

[0083] In addition, the terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include at least one of the features. In the representation of the present disclosure, the meaning of "plurality" is two or more, unless otherwise clearly and specifically defined.

[0084] In order to clearly describe the present disclosure, components not related to the description are omitted, and the same reference numerals are given to the same or similar components throughout the specification.

[0085] Throughout the specification, when a device is said to be "connected" to another device, this includes not only the case of "direct connection" but also the case of "indirect connection" by placing other elements therebetween. In addition, when a device is said to "include" a certain component, unless otherwise stated, it does not exclude other components, but means that other components may be included.

[0086] When a device is said to be "on" another device, it may be directly on the other device, but there may also be other devices between it. In contrast, when a device is said to be "directly" on another device, there are no other devices between it.

[0087] Although the terms first, second, etc. are used to represent various elements in some examples, these elements should not be limited by these terms. These terms are only used to distinguish one element from another element. For example, the first interface and the second interface, etc. are represented. Moreover, as used in this article, the singular forms "one", "one" and "the" are intended to also include plural forms, unless there is an opposite indication in the context. It should be further understood that the terms "comprising" and "including" indicate the existence of features, steps, operations, elements, components, projects, kinds, and / or groups, but do not exclude the existence, occurrence or addition of one or more other features, steps, operations, elements, components, projects, kinds, and / or groups. The terms "or" and "and / or" used herein are interpreted as inclusive, or mean any one or any combination. Therefore, "A, B or C" or "A, B and / or C" means "any of the following: A; B; C; A and B; A and C; B and C; A, B and C". Only when the combination of elements, functions, steps or operations is inherently mutually exclusive in some way, will there be an exception to this definition.

[0088] The technical terms used herein are only used to refer to specific embodiments and are not intended to limit the present disclosure. The singular form used herein also includes the plural form unless the sentence clearly indicates the contrary meaning. The meaning of "including" used in the specification is to specify specific characteristics, regions, integers, steps, operations, elements and / or components, and does not exclude the existence or addition of other characteristics, regions, integers, steps, operations, elements and / or components.

[0089] Although not defined differently, all terms, including technical and scientific terms used herein, have the same meaning as those generally understood by those skilled in the art to which the present disclosure belongs. Terms defined in commonly used dictionaries are additionally interpreted as having meanings consistent with the content of relevant technical literature and current disclosures, and unless defined, shall not be overly interpreted as ideal or very formal meanings.

[0090] According to the relevant explanations and descriptions in the prior art, in order to reduce the amount of calculation of the theoretical spectral data calculation method and improve the calculation efficiency of the theoretical spectral data, the technical solution provided by the present disclosure provides a theoretical spectral data calculation method. The specific implementation of this calculation method will be specifically explained below:

[0091] Specifically, in some embodiments provided in the present disclosure, Figure 1 A schematic diagram of the process of calculating theoretical spectral data is provided. Figure 1 As shown, the calculation method may specifically include the following steps:

[0092] Step 100: Obtain a coupled wave equation group in differential form for a sample having a periodic structure in both the first direction and the second direction according to a rigorous coupled wave analysis algorithm.

[0093] Step 200: Integrate the coupled wave equations in the height direction of the periodic structure to obtain an exponential coupled wave equation, wherein the height direction, the first direction and the second direction are perpendicular to each other, and the exponential coupled wave equation includes a matrix index.

[0094] Step 300: Obtain a matrix in the matrix exponential of the coupled wave equations in exponential form as the first matrix.

[0095] Step 400: Convert the matrix index to the product of the second matrix index and the third matrix index raised to the power of b, so that the denominators of the fractions included in the matrices in the second matrix index and the third matrix index both include b, where b is a preset integer and b≥2.

[0096] Step 500: Convert the exponent of the second matrix into the first block matrix, convert the exponent of the third matrix into the second block matrix, and obtain the calculation formula corresponding to the matrix exponent based on the result of matrix diagonalization of the matrices common to the blocks of the first block matrix and the result of matrix diagonalization of the matrices common to the blocks of the second block matrix, wherein the dimension of the matrix in each block is half of the dimension of the first matrix.

[0097] Step 600: Obtain the reflection coefficient corresponding to the periodic structure according to the calculation formula corresponding to the matrix index and the electromagnetic field continuity condition of the periodic structure surface.

[0098] Step 700: Obtain theoretical spectrum data corresponding to the periodic structure according to the reflection coefficient.

[0099] The theoretical spectral data may include a reflectance spectrum, a spectrum of polarization state changes, a spectrum of Fourier coefficients of polarization state analysis, or a spectrum directly output to describe the scattering process. The spectrum directly output to describe the scattering process may be, for example, an NCS spectrum or a Mueller spectrum, etc., which is not limited here.

[0100] It can be understood that through the above steps 100 to 700, the optimization of the calculation of theoretical spectral data can be achieved. By converting the matrix exponents in the coupled wave equations in exponential form into corresponding calculation formulas, the matrix dimension that needs to be diagonalized is reduced by half compared with the dimension of the first matrix, thereby improving the speed of matrix diagonalization and thus improving the calculation efficiency of theoretical spectral data. In the embodiments of the present disclosure, the specific implementation of the above steps 100 to 700 will be further described below.

[0101] In the above embodiment, it can be understood that before the aforementioned step 100 is performed, it is necessary to obtain the structural parameters and optical parameters of the sample with a periodic structure in advance. Among them, the structural parameters are the morphological parameters of the periodic structure, and the structural parameters used here are known parameters of the periodic structure in terms of morphology, such as the period of the periodic structure in the first direction and the second direction; the optical parameters may include the dielectric constant of the periodic structure, the wavelength of the incident light in the optical measurement, the angle of the incident light, etc. How to obtain the structural parameters and optical parameters are all existing technologies and will not be repeated here.

[0102] In the above embodiment, in the specific implementation of the above step 100, Figure 2 A schematic flow chart of obtaining the coupled wave equations in differential form is provided, such as Figure 2 As shown, the content in step 100 may specifically include the following steps 101 and 102.

[0103] Step 101: Perform Fourier series expansion on the magnetic field, electric field and dielectric constant corresponding to the periodic structure. Figure 3 FIG. 4 shows a schematic cross-sectional view of a sample having a periodic structure in both a first direction X and a second direction Y in the height direction. Figure 3 As shown in the figure, the sample includes a substrate 001 and a periodic structure 002 arranged on the surface of the substrate 001. The periodic structure 002 is periodically repeated in the first direction X and the second direction Y. Figure 3 Taking the sample with periodic structure shown as an example, the specific implementation of Fourier series expansion of magnetic field, electric field and dielectric constant is explained.

[0104] In the above step 101, the mathematical expressions of Fourier series expansion of the electric field at position z in the height direction Z in the first direction X and the second direction Y are as follows:

[0105]

[0106] Among them, E x (z) is the component of the electric field at position z in the first direction X, E y (z) is the component of the electric field at position z in the second direction Y; j is an imaginary unit; M is the cutoff order in the first direction X, and N is the cutoff order in the second direction Y; k xm Characterizes the wave vector component in the first direction X, at position x and with diffraction order m, k yn Characterizes the wave vector component in the second direction Y, at position y and with diffraction order n. Those skilled in the art know that the above two wave vector components contain information about the period of the periodic structure in the first direction X, the period in the second direction Y and the angle of the incident light; Sx，mm (z) is the electric field component E at position z x (z) The corresponding Fourier coefficient, S y，mm (z) is the electric field component E at position z y (z) The Fourier coefficient corresponding to the first direction X, where x is the position in the first direction X, and y is the position in the second direction Y.

[0107] In the above step 101, the mathematical expressions of Fourier series expansion of the magnetic field at position z in the first direction X and the second direction Y are as follows:

[0108]

[0109] Among them, H x (z) is the component of the magnetic field at position z in the first direction X, H y (z) is the component of the magnetic field at position z in the second direction Y; ε0 is the vacuum dielectric constant; μ0 is the vacuum magnetic permeability; U x，mm (z) is the magnetic field component H at position z x (z) The corresponding Fourier coefficient, U y，mm (z) is the magnetic field component H at position z y (z) The corresponding Fourier coefficients.

[0110] In the above step 101, the mathematical expression of the Fourier series expansion of the dielectric constant is as follows:

[0111]

[0112] Among them, Λ x is the period of the periodic structure in the first direction X, Λ y is the period of the periodic structure in the second direction Y; ε(x, y, λ) is the dielectric constant of the periodic structure, which is related to the wavelength of the incident light and is a function of the wavelength of the incident light; ε mn (λ) is the Fourier coefficient corresponding to the dielectric constant of the periodic structure with a diffraction order of m in the first direction X and a diffraction order of n in the second direction Y.

[0113] Step 102: Substitute the obtained Fourier series expansions of the magnetic field, electric field and dielectric constant into the Maxwell equations to obtain the coupled wave equations in differential form. Specifically, the mathematical expression of the coupled wave equations in differential form is as follows:

[0114]

[0115] Where k0 is the wave vector in vacuum environment; P and Q are the optical property matrices corresponding to the sample; S x(z) represents the Fourier coefficient matrix corresponding to the component of the electric field in the first direction X at position z, where the element in the mth row and nth column is the component E of the electric field mentioned above x (z) The corresponding Fourier coefficient S x，mn (z); S y (z) represents the Fourier coefficient matrix corresponding to the component of the electric field in the second direction Y at position z, where the element in the mth row and nth column is the component E of the electric field mentioned above y (z) The corresponding Fourier coefficient S y，mn (z); U x (z) represents the Fourier coefficient matrix corresponding to the component of the magnetic field in the first direction X at position z, where the element in the mth row and nth column is the component H of the magnetic field mentioned above x (z) The corresponding Fourier coefficient U x，mn (z); U y (z) represents the Fourier coefficient matrix corresponding to the component of the magnetic field in the second direction Y at position z, where the element in the mth row and nth column is the component H of the magnetic field mentioned above y (z) The corresponding Fourier coefficient U y，mn (z); z represents the position of the periodic structure in the height direction.

[0116] It can be understood that based on the above formulas 1 to 4, it can be seen that for the Fourier coefficient matrix of the electric field or magnetic field in the first direction X or the second direction Y, that is, for the Fourier coefficient matrix S x (z), S y (z), U x (z), U y (z), the corresponding dimension is (2M+1)(2N+1), where M is the truncation order of the Fourier series expansion in the first direction, and N is the truncation order of the Fourier series expansion in the second direction; then for the matrix Its dimension is 4(2M+1)(2N+1). Accordingly, in order to satisfy the matrix multiplication condition, the matrix The dimension of is also 4(2M+1)(2N+1), and the dimensions of the corresponding optical property matrices P and Q are 2(2M+1)(2N+1) respectively. The elements in the above optical property matrices P and Q are related to optical parameters and can also be called optical parameter matrices, which will not be described in detail here.

[0117] In the above embodiment, in the specific implementation of the above step 200, in the process of integrating the above differential coupled wave equations to obtain the exponential coupled wave equations, in an optional specific implementation of the embodiment of the present disclosure, as Figure 3As shown, the coupled wave equations in differential form can be integrated from the lower surface height of the periodic structure 002 to the upper surface height of the periodic structure 002 in the height direction Z, thereby obtaining the coupled wave equations in exponential form. Figure 3 Taking the three-dimensional rectangular coordinate system shown as an example, the XOY plane formed by the first direction X, the second direction Y and the origin O is coplanar with the lower surface of the periodic structure 002. When H is set as the distance from the lower surface of the periodic structure to the upper surface of the periodic structure in the height direction Z, the position z=0 corresponds to the lower surface of the periodic structure 002, and the position z=H corresponds to the upper surface of the periodic structure 002. In another optional specific implementation of the embodiment of the present disclosure, the XOY plane formed by the first direction X, the second direction Y and the origin O can also be set to be coplanar with the upper surface of the periodic structure 002, which is not limited here.

[0118] In the above embodiment, specifically, when the XOY plane formed by the first direction X and the second direction Y and the origin O is coplanar with the lower surface of the periodic structure 002, the mathematical expression of the coupled wave equations in exponential form can be as follows:

[0119]

[0120] Wherein, 0≤z≤H, H is the distance from the lower surface of the periodic structure to the upper surface of the periodic structure in the height direction, and exp is the natural base of the matrix index. Accordingly, the mathematical expression of the matrix index included therein can be shown as follows:

[0121]

[0122] Accordingly, the mathematical expression of the first matrix in obtaining the matrix index in the aforementioned step 300 can be as follows:

[0123]

[0124] It can be understood that, since the dimensions of the optical characteristic matrices P and Q are 2 (2M+1) (2N+1) respectively, the dimension of the first matrix can be obtained to be 4 (2M+1) (2N+1). In the prior art, the coupled wave equations in exponential form are first obtained based on the coupled wave equations in differential form, and then in the process of calculating the coupled wave equations in exponential form, the above-mentioned first matrix is ​​directly matrix diagonalized, and then the index of the first matrix (in this embodiment, the index of the first matrix is ​​called the matrix index, and the index of the first matrix is ​​the matrix index in step 200 and step 300) corresponding to the calculation formula is obtained. Since the dimension of the first matrix reaches 4 (2M+1) (2N+1), the amount of calculation of the matrix diagonalization is large, which takes up a lot of time, which is not conducive to improving the calculation efficiency of the theoretical spectral data. Therefore, in the embodiment provided in the present disclosure, the first matrix is ​​split and transformed to reduce the amount of calculation, thereby achieving the improvement of the calculation efficiency of the theoretical spectral data. The specific implementation of the splitting and transformation of the first matrix will be described in detail below.

[0125] In the above embodiment, in the specific implementation of the above step 400, Figure 4 A schematic diagram of a process for converting a matrix index into the product of the index of the second matrix and the index of the third matrix to the power of b is shown. Figure 4 As shown, the content in step 400 may specifically include the following steps 401 to 403.

[0126] Step 401: Convert the first matrix into the sum of a symmetric matrix and an antisymmetric matrix. Specifically, the mathematical expression of the symmetric matrix can be shown as follows:

[0127]

[0128] The mathematical expression of an antisymmetric matrix can be shown as follows:

[0129]

[0130] Step 402: Obtain a second matrix and a third matrix according to the symmetric matrix and the antisymmetric matrix, wherein the second matrix is ​​the quotient of the symmetric matrix and b, and the third matrix is ​​the quotient of the antisymmetric matrix and b. Specifically, the mathematical expression of the second matrix can be as follows:

[0131]

[0132] The mathematical expression of the third matrix can be shown as follows:

[0133]

[0134] Step 403: Obtain the b-th power of the product based on the exponent of the second matrix and the exponent of the third matrix. Specifically, the mathematical expression of the b-th power of the product can be as follows:

[0135]

[0136] It is understandable that in matrix-related exponential operations, such as e B+C =e B e C The mathematical expression of cannot be directly established, where B and C are both matrices; however, the matrix with larger values ​​on the left side of the equation can be split into matrices with smaller values, and then the product of the matrices with smaller values ​​can be used to approximate the matrix with larger values, that is, the mathematical expression shown below is established:

[0137] e B+C =lim b→∞ (e B / b e C / b ) b (Formula 15);

[0138] Wherein b is an integer. If b is an integer between [2, 1000], the above formula 15 can be further expressed as follows:

[0139] e B+C ≈(e B / b e C / b ) b (Formula 16).

[0140] Therefore, in the above steps 401 to 403, the above matrix splitting method is applied to the relevant embodiments of the present disclosure, and the first matrix corresponding to the matrix index is first split into the sum of a symmetric matrix and an antisymmetric matrix, and then the corresponding second matrix and third matrix are obtained according to the quotient of the symmetric matrix and the antisymmetric matrix with b, and finally the matrix index, the index of the second matrix and the index of the third matrix are respectively brought into the corresponding positions of Formula 16, and the approximate equation shown below can be obtained:

[0141]

[0142] Thereby, a matrix splitting process is realized in which the matrix exponent is converted into the b-th power of the product of the exponent of the second matrix and the exponent of the third matrix.

[0143] In the above embodiment, in the specific implementation of the above step 500, Figure 5 A schematic diagram of a flow chart for obtaining a calculation formula corresponding to a matrix index is shown. Figure 5 As shown, the content in step 500 may specifically include the following steps 501 to 505.

[0144] Step 501: Convert the exponent of the second matrix into a first block matrix, so that each block in the first block matrix is ​​a hyperbolic function of the fourth matrix. Specifically, the mathematical expression for converting the exponent of the second matrix into the first block matrix is ​​as follows:

[0145]

[0146] in, is the fourth matrix, is the hyperbolic sine function corresponding to the fourth matrix, is the hyperbolic cosine function corresponding to the fourth matrix, and the fourth matrix is ​​the common matrix in each block of the first block matrix.

[0147] Step 502: Convert the exponent of the third matrix into a second block matrix, so that each block in the second block matrix is ​​a trigonometric function of the fifth matrix. Specifically, the mathematical expression for converting the exponent of the third matrix into the second block matrix is ​​as follows:

[0148]

[0149] in, is the fifth matrix, is the cosine function corresponding to the fifth matrix, is the sine function corresponding to the fifth matrix, and the fifth matrix is ​​the common matrix in each block of the second block matrix.

[0150] It is understandable that the above steps 501 and 502 may be performed synchronously or asynchronously, which is not limited here.

[0151] Step 503: Diagonalize the fourth matrix and the fifth matrix respectively, that is, implement matrix diagonalization respectively. Specifically, the mathematical expressions corresponding to the diagonalization of the fourth matrix and the fifth matrix are respectively as follows:

[0152]

[0153] Among them, D1 and D2 are diagonal matrices, U1 and U2 are orthogonal matrices, and I is the unit matrix.

[0154] Step 504: Based on the result of the diagonalization of the fourth matrix (specifically ) obtains the first calculation formula corresponding to the exponent of the second matrix, based on the result of the diagonalization of the fifth matrix (specifically ) obtains the second calculation formula corresponding to the exponent of the third matrix. Specifically, the corresponding mathematical expressions are as follows:

[0155]

[0156]

[0157] Among them, the first calculation formula is:

[0158]

[0159] The second calculation formula is:

[0160]

[0161] Step 505: Convert the matrix index to the product of the first calculation formula and the second calculation formula to the power of b, so as to obtain the calculation formula corresponding to the matrix index. Specifically, the corresponding mathematical expression is as follows:

[0162]

[0163] Among them, the calculation formula corresponding to the matrix index is:

[0164]

[0165] Among them, sh(D1) is the hyperbolic sine function corresponding to the diagonal matrix D1, ch(D1) is the hyperbolic cosine function corresponding to the diagonal matrix D, cos(D2) is the cosine function corresponding to the diagonal matrix D2, and sin(D2) is the sine function corresponding to the diagonal matrix D2.

[0166] It can be understood that in the above steps 501 to 505, since the dimensions of the optical property matrices P and Q are 2(2M+1)(2N+1) respectively, the corresponding dimensions of the fourth matrix and the fifth matrix are both 2(2M+1)(2N+1), that is, the dimensions of the fourth matrix and the fifth matrix are both half of the dimensions of the first matrix. In the process of diagonalizing the fourth matrix and the fifth matrix, the amount of calculation is less and the speed is faster than that of diagonalizing the first matrix, thereby achieving an improvement in the calculation efficiency of theoretical spectral data.

[0167] In the above embodiment, in the specific implementation of the above step 600, Figure 6 FIG. 1 shows a schematic diagram of a process for obtaining the reflection coefficient corresponding to a periodic structure. Figure 6 As shown, the content in step 600 may specifically include the following steps 601-602.

[0168] Step 601: Substitute the calculation formula corresponding to the matrix index into the coupled wave equations in exponential form.

[0169] Step 602: Obtain the reflection coefficient according to the coupled wave equations in exponential form after substitution and the electromagnetic field continuity conditions at the upper surface position and the lower surface position of the periodic structure in the height direction. The specific implementation of the above step 602 belongs to the common knowledge of those skilled in the art. In addition, the content in step 700 is also common knowledge and will not be repeated here.

[0170] In summary, through the technical solution provided by the present invention, in the process of calculating theoretical spectral data, for the matrix index in the coupled wave equation group in exponential form, the matrix index is first converted into the bth power of the product of the index of the second matrix and the index of the third matrix, and then the index of the second matrix and the index of the third matrix are respectively converted into block matrices, and the matrices in the blocks are used to implement matrix diagonalization, so as to convert the matrix index into the corresponding calculation formula; compared with the prior art of directly performing matrix diagonalization on the first matrix in the matrix index and then substituting the diagonalization result into the matrix index to obtain the calculation formula corresponding to the matrix index, in the theoretical spectral data calculation method provided by the present invention, since the matrix dimension required for matrix diagonalization is reduced, the speed of matrix diagonalization is improved, a large amount of calculation is saved, and thus the calculation efficiency of theoretical spectral data is improved.

[0171] The above content is a further detailed description of the present disclosure in combination with specific preferred implementation modes, and it cannot be determined that the specific implementation of the present disclosure is limited to these descriptions. For ordinary technicians in the technical field to which the present disclosure belongs, several simple deductions or substitutions can be made without departing from the concept of the present disclosure, which should be regarded as falling within the scope of protection of the present disclosure.

Claims

1. A method for calculating theoretical spectral data, characterized in that: The steps include: Obtaining a coupled wave equation set in differential form for a sample having a periodic structure in both a first direction and a second direction according to a rigorous coupled wave analysis algorithm; Integrating the coupled wave equations in the height direction of the periodic structure to obtain an exponential coupled wave equation, wherein the height direction, the first direction and the second direction are perpendicular to each other, and the exponential coupled wave equation includes a matrix index; Obtaining a matrix in the matrix index as a first matrix; Converting the matrix index into the product of the index of the second matrix and the index of the third matrix raised to the power of b, so that the denominators of the fractions included in the matrices in the index of the second matrix and the index of the third matrix both include b, where b is a preset integer and b≥2; Converting the exponents of the second matrix into the first block matrix, converting the exponents of the third matrix into the second block matrix, and obtaining calculation formulas corresponding to the matrix exponents based on the result of matrix diagonalization of the matrices common to the blocks of the first block matrix and the result of matrix diagonalization of the matrices common to the blocks of the second block matrix, wherein the dimensions of the matrices in each of the blocks are half of the dimensions of the first matrix; Obtaining a reflection coefficient corresponding to the periodic structure according to the calculation formula and the electromagnetic field continuity condition on the surface of the periodic structure; Theoretical spectrum data corresponding to the periodic structure is obtained according to the reflection coefficient.

2. The method for calculating theoretical spectrum data according to claim 1, characterized in that: The step of integrating the coupled wave equations in the height direction of the periodic structure to obtain an exponential coupled wave equation, comprises the following steps: The coupled wave equations in differential form are integrated from the lower surface height of the periodic structure to the upper surface height of the periodic structure in the height direction of the periodic structure to obtain the coupled wave equations in exponential form.

3. The method for calculating theoretical spectrum data according to claim 1, characterized in that: The step of converting the exponents of the second matrix into the first block matrix and converting the exponents of the third matrix into the second block matrix, obtaining the calculation formulas corresponding to the matrix exponents based on the result of matrix diagonalization of the matrices common to the blocks of the first block matrix and the result of matrix diagonalization of the matrices common to the blocks of the second block matrix, wherein the dimensions of the matrices in each of the blocks are half of the dimensions of the first matrix, comprises the following steps: The exponents of the second matrix are converted into a first block matrix, so that each block in the first block matrix is ​​a hyperbolic function of the fourth matrix, and the exponents of the third matrix are converted into a second block matrix, so that each block in the second block matrix is ​​a trigonometric function of the fifth matrix, and the calculation formulas corresponding to the matrix exponents are obtained based on the results of matrix diagonalization performed on the fourth matrix and the fifth matrix, respectively, wherein the dimensions of the fourth matrix and the fifth matrix are both half of the dimensions of the first matrix.

4. The method for calculating theoretical spectrum data according to claim 3, characterized in that: The method of obtaining a coupled wave equation group in differential form of a sample having a periodic structure in both the first direction and the second direction according to a rigorous coupled wave analysis algorithm comprises the following steps: Performing Fourier series expansion on the magnetic field, electric field and dielectric constant corresponding to the periodic structure respectively; Substituting the obtained Fourier series expansion of the magnetic field, the electric field and the dielectric constant into the Maxwell equations to obtain the coupled wave equations in differential form; The mathematical expression of the coupled wave equations in differential form is as follows: Among them, k0 is the wave vector in vacuum environment, P and Q are the optical property matrices corresponding to the sample, and S x (z) represents the Fourier coefficient matrix corresponding to the component of the electric field in the first direction at position z, S y (z) represents the Fourier coefficient matrix corresponding to the component of the electric field in the second direction at position z, U x (z) represents the Fourier coefficient matrix corresponding to the component of the magnetic field in the first direction at position z, U y (z) represents the Fourier coefficient matrix corresponding to the component of the magnetic field in the second direction at position z, and z represents the position of the periodic structure in the height direction.

5. The method for calculating theoretical spectrum data according to claim 4, characterized in that: The mathematical expression of the exponential form of the coupled wave equations is as follows: Wherein, 0≤z≤H, H is the distance from the lower surface of the periodic structure to the upper surface of the periodic structure in the height direction; The mathematical expression of the first matrix is ​​as follows: Among them, the dimension of the first matrix is ​​4(2M+1)(2N+1), M is the truncation order of the Fourier series expansion in the first direction, N is the truncation order of the Fourier series expansion in the second direction, and exp is the natural base of the matrix exponent.

6. The method for calculating theoretical spectrum data according to claim 5, characterized in that: The step of converting the matrix index into the product of the index of the second matrix and the index of the third matrix raised to the power of b, so that the denominators of the fractions included in the matrices in the index of the second matrix and the index of the third matrix both include b, comprises the following steps: The first matrix is ​​converted into the sum of a symmetric matrix and an antisymmetric matrix, wherein the mathematical expression of the symmetric matrix is ​​as follows: The mathematical expression of the antisymmetric matrix is ​​as follows: Obtaining the second matrix and the third matrix according to the symmetric matrix and the antisymmetric matrix, wherein the second matrix is ​​the quotient of the symmetric matrix and b, and the third matrix is ​​the quotient of the antisymmetric matrix and b; The mathematical expression of the second matrix is: The mathematical expression of the third matrix is: The b-th power of the product is obtained based on the exponent of the second matrix and the exponent of the third matrix. The corresponding mathematical expression is:

7. The method for calculating theoretical spectrum data according to claim 6, characterized in that: The mathematical expression for converting the exponent of the second matrix into the first block matrix is ​​as follows: in, is the fourth matrix, is the hyperbolic sine function corresponding to the fourth matrix, is the hyperbolic cosine function corresponding to the fourth matrix; The mathematical expression for converting the exponent of the third matrix into the second block matrix is ​​as follows: in, is the fifth matrix, is the cosine function corresponding to the fifth matrix, is the sine function corresponding to the fifth matrix.

8. The method for calculating theoretical spectrum data according to claim 7, characterized in that: The dimensions of the fourth matrix and the fifth matrix are both 2(2M+1)(2N+1).

9. The method for calculating theoretical spectrum data according to claim 7, characterized in that: The step of obtaining the calculation formula corresponding to the matrix index based on the result of respectively performing matrix diagonalization on the fourth matrix and the fifth matrix comprises the following steps: The fourth matrix and the fifth matrix are diagonalized respectively, and the mathematical expressions corresponding to the diagonalization of the fourth matrix and the fifth matrix are respectively as follows: Among them, D1 and D2 are diagonal matrices, U1 and U2 are orthogonal matrices, and I is the unit matrix; The first calculation formula corresponding to the exponent of the second matrix is ​​obtained based on the result of the diagonalization of the fourth matrix, and the second calculation formula corresponding to the exponent of the third matrix is ​​obtained based on the result of the diagonalization of the fifth matrix. The corresponding mathematical expressions are respectively as follows: Wherein, the first calculation formula is: The second calculation formula is: The matrix index is converted into the bth power of the product of the first calculation formula and the second calculation formula to obtain the calculation formula corresponding to the matrix index. The corresponding mathematical expression is as follows: Among them, the calculation formula corresponding to the matrix index is: Among them, sh(D1) is the hyperbolic sine function corresponding to the diagonal matrix D1, ch(D1) is the hyperbolic cosine function corresponding to the diagonal matrix D1, cos(D2) is the cosine function corresponding to the diagonal matrix D2, and sin(D2) is the sine function corresponding to the diagonal matrix D2.

10. The method for calculating theoretical spectrum data according to claim 1, characterized in that: The step of obtaining the reflection coefficient corresponding to the periodic structure according to the calculation formula and the electromagnetic field continuity condition on the surface of the periodic structure comprises the following steps: Substituting the calculation formula corresponding to the matrix index into the coupled wave equations in the exponential form; The reflection coefficient is obtained based on the exponential form of the coupled wave equations, the electromagnetic field continuity conditions at z=0 and z=H, wherein z=0 represents the upper surface position or the lower surface position of the periodic structure in the height direction; z=H represents the lower surface position or the upper surface position of the periodic structure in the height direction.

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