An ellipsometric parameter calculation method based on mueller matrix dimension reduction

Abstract

The application discloses a Mueller matrix dimension reduction-based ellipsometric parameter calculation method, which comprises the following steps: testing the Mueller matrix of a scattering substance; calculating the eigenvalue of a corresponding coherent matrix of the Mueller matrix of the scattering substance and analyzing the eigenvalue; constructing a dimension-reduced polarization space and a depolarization space of the scattering substance; optimizing decomposition parameters, establishing a decomposed physical equation, and adopting an optimization method to perform ellipsometric parameter calculation and noise analysis. The application can process the Mueller matrix dimension reduction and ellipsometric parameter calculation of rough substances and film samples, and has the advantages of low noise, small calculation amount, strong practicability and high precision.

Description

Technical Field

[0001] This invention belongs to the field of polarization technology, specifically a method for calculating ellipticity parameters based on Mueller matrix dimensionality reduction. Background Technology

[0002] Polarization measurement technology is an important optical measurement technology. In recent years, it has been widely used in fields such as semiconductor materials, biological tissue analysis, medical testing and diagnosis, radar target recognition, and remote sensing target recognition.

[0003] With the rapid development of the semiconductor and microelectronics industries, polarization measurement technology has seen rapid advancements in fields such as semiconductors and thin film materials. Ellipsometry parameters are crucial physical parameters for thin films, semiconductor materials, and crystal materials, and can be used to calculate parameters such as film thickness and refractive index, as well as monitor film quality. Ellipsometry measurements utilize the principle of Fresnel reflection, and for smooth, ideal media, measurements can be directly performed using an ellipsometer. However, in practical research, for rough materials or non-uniform thin film samples, the uneven surface and severe scattering make direct testing with an ellipsometer difficult.

[0004] Mueller polarization measurement techniques can test the Mueller matrix of rough media. The 16 elements of the Mueller matrix contain all polarization and depolarization information of the target medium. Because the Mueller matrix involves complex calculations with 16 degrees of freedom and multiple parameters, and the ellipsometric parameter is related to polarization information, it is necessary to remove depolarization information caused by scattering. The calculation of the ellipsometric parameter requires dimensionality reduction decomposition of the Mueller matrix, while simultaneously reducing errors and noise during the experiment. Summary of the Invention

[0005] To address the technical problems mentioned in the background section, this invention proposes a method for calculating elliptic parameters based on Mueller matrix dimensionality reduction.

[0006] The technical solution to achieve the objective of this invention is as follows: Firstly, this invention provides a method for calculating elliptic deviation parameters based on Mueller matrix dimensionality reduction, comprising the following steps:

[0007] The Mueller matrix of the scattering material was tested.

[0008] Calculate and analyze the eigenvalues ​​of the coherence matrix corresponding to the Mueller matrix of the scattering material;

[0009] Construct a dimension-reduced polarization space and a depolarization space for scattering matter;

[0010] Optimize the decomposition parameters, establish the physical equations for the decomposition, and use optimization methods to calculate elliptic parameters and perform noise analysis.

[0011] Furthermore, the polarization space is represented by ellipticity parameters. express, The amplitude ratio of the s and p components in the emitted light is represented by Δ, and the phase difference between the s and p components is represented by Δ.

[0012]

[0013] In the formula r s r p ρ represents the total reflectance of s and p light, respectively. s , ρ p φ represents the amplitude of s and p light. s φ p These represent the phases of the s and p beams, respectively.

[0014] Furthermore, a Mueller polarization imager was used as the testing system, and the expression for the Mueller matrix is ​​as follows:

[0015] S' = MS

[0016] Where S' is the Stokes vector of the outgoing light, S is the Stokes vector of the incident light, and M is the Mueller matrix of the rough surface material.

[0017] Furthermore, in step 2, the coherence matrix of the scattering matrix is ​​calculated as follows:

[0018]

[0019] Among them, M ij For the elements of the Mueller matrix, σ i ,σ j All are Pauli matrices.

[0020] Furthermore, the analysis of eigenvalues ​​includes physical realizability analysis, determining whether the eigenvalues ​​of the coherence matrix are greater than or equal to 0. If any eigenvalue is less than 0 or a specific threshold, it indicates that the measurement of the Mueller matrix has noise and error, and the eigenvalue needs to be set to 0. Simultaneously, eigenvalue normalization and Mueller matrix reconstruction are performed, and the calculation formula is as follows:

[0021]

[0022] In the formula, λ i Let represent the eigenvalues ​​of the coherence matrix, i = 1, 2, 3, tr represent the trace of the matrix, and M represent the eigenvalues ​​of the coherence matrix. i This represents the submatrix of the Mueller matrix decomposition.

[0023] Furthermore, the 4×4 complex space is transformed into Euclidean space using dimensionality reduction methods:

[0024]

[0025] In the formula, U 4RLet U3 be a complex vector unitary matrix, a 3×3 submatrix; t denotes the transpose operation. U3 contains a complex phase factor, which is ignored during the transformation. Its three eigenvalues ​​remain unchanged in Euclidean space. According to group theory, U3 consists of three submatrices with six parameters. During the depolarization process caused by scattering, due to the disappearance of spatial symmetry, the eigenvectors of U3 can be ignored, and only its eigenvalues ​​need to be considered. Therefore, U4 consists of one eigenvalue λ1 and two eigenvector parameters (a, δ). The eigenvector corresponding to the eigenvalue λ1 has two parameters and corresponds to a diagonal matrix. This diagonal matrix form is...

[0026]

[0027] In the formula, a and δ represent the angular parameters of the polarization part in scattering, which depend on factors such as phase angle, surface material composition, and particle size, and U4 is a unitary matrix.

[0028] U3 includes only 3 eigenvalues ​​λ i The target has (i = 2, 3, 4) and 0 eigenvectors, therefore the coherence matrix of the target is represented as:

[0029]

[0030] The Mueller matrix in Euclidean space is decomposed into

[0031] M=(1-d)M J +M D

[0032] In the formula,

[0033]

[0034] d represents the depolarization coefficient, M D Represents the depolarization matrix related to scattering:

[0035]

[0036] Its ellipticity parameters can be calculated using the least squares method.

[0037]

[0038] In the formula, ||.|| represents the 2-norm. Using the least squares method can improve the decomposition accuracy and reduce the error, thus calculating the polarization parameters.

[0039] In a second aspect, the present invention provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the method described in the first aspect.

[0040] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method described in the first aspect.

[0041] Fourthly, the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the method described in the first aspect.

[0042] Compared with existing technologies, the advantages of this invention are as follows: This invention reduces the dimensionality of the high-dimensional, multi-parameter Mueller matrix, transforming the solution of multiple parameters into the calculation of two elliptic polarization parameters; simultaneously, it uses an optimization method to calculate the ellipticity parameters of the thin film sample, which is difficult to achieve using traditional ellipsometer measurement methods; furthermore, the decomposition accuracy is significantly improved, reducing experimental errors. It is more practical, effective, accurate, precise, and widely applicable than traditional testing methods. Attached Figure Description

[0043] Figure 1 This is a flowchart of the method of the present invention.

[0044] Figure 2 This is a schematic diagram of the experimental setup.

[0045] Figure 3 This is a schematic diagram of a camouflaged sample film.

[0046] Figure 4 This is a schematic diagram of elliptic parameters.

[0047] Figure 5 This is a comparative diagram of the LSM method and the Cloude method of this invention. Detailed Implementation

[0048] This invention proposes a method for calculating elliptic partial parameters based on Mueller matrix dimensionality reduction, comprising the following steps:

[0049] (1) Test the Mueller matrix of the scattering material;

[0050] Specifically, a Mueller polarization imager was used as the testing system.

[0051] The expression for the Mueller matrix is ​​as follows:

[0052] S' = MS

[0053] Where S' is the Stokes vector of the outgoing light, S is the Stokes vector of the incident light, and M is the Mueller matrix of the rough surface material.

[0054] (2) Calculate and analyze the eigenvalues ​​of the coherence matrix corresponding to the Mueller matrix of the scattering material;

[0055] Further analysis of eigenvalues ​​includes noise analysis and physical reliability analysis.

[0056] (3) The Mueller matrix coherence matrix is ​​decomposed into two parts, and the seven parameters of the unitary matrix space of the polarization part are transformed into two parameters of the Euclidean space.

[0057] (4) Calculate the ellipsoid parameters of the scattering material;

[0058] (5) Establish the physical equations for decomposition, conduct error analysis, and evaluate accuracy.

[0059] Furthermore, in step (3), the polarization matrix parameters of the Mueller matrix of the rough material are calculated as follows:

[0060] The polarization and depolarization problems generated can be constructed using the eigenvectors of e1. First, construct the U4 matrix.

[0061]

[0062] In the formula, α, β, and γ are parameterized angles, and δ i The phase information parameters are i = 1, 2, 3, 4.

[0063]

[0064] Among them, the eigenvectors e in the unitary matrix i (i = 1, 2, 3, 4)

[0065]

[0066] A unitary matrix consists of four eigenvectors and seven unknown parameters: α, β, γ, and δ. i i = 1, 2, 3, 4, including complex phase operations e of complex numbers. iδ .

[0067] A matrix can be transformed into

[0068]

[0069] This formula determines the depolarization generated by U3 during the scattering process in the target medium. The operation of U3 involves a series of continuous and smooth transformations to generate the polarization-deviated portion, i.e., realizing λ in the coherence matrix. i Separation of (i=2,3,4) from λ1.

[0070] The parameters and eigenvalues ​​of the coherence matrix of the target Mueller matrix satisfy the following equation:

[0071] T = (E + K) + [E + K)

[0072] In the formula, (...) represents the polarization part, [...] represents the depolarization part, E represents the parameters of the eigenvector, and K represents the number of parameters of the eigenvalue.

[0073] Matrix dimensionality reduction works by considering the relationship between parameters and dimension in the matrix space as follows:

[0074] (K) + [K] = 4

[0075] (E)+[E]=dim(SU(4))-rank(SU(4))=12

[0076] Here, SU(4) represents a combination of unitary matrices, that is, a group of unitary matrices with an identity determinant. According to group theory, we need 15 parameters for parameterization. Therefore, to reduce dimensionality, we need ground states represented by a lower-order group; SU(3) requires 8, and SU(2) requires 2. In general, a 3×3 unitary matrix U3 also has a diagonal phase shift matrix. This operation adds three degrees of freedom parameters. However, if we further consider that the unitary matrix satisfies det(U3) = 1, only two independent phase angles and eight degrees of freedom parameters are obtained from D3 in SU(3). However, we are only interested in the Hermitian coherence matrix formed, so D3 is hidden in the construction of the depolarization space. In this case, only the eigenvalues ​​λ of the coherence matrix related to scattering need to be considered. i (i = 2, 3, 4), without needing to calculate the phase of the vector, as follows:

[0077]

[0078] In the formula φ i (i = 1, 2, 3), δ i (i = 1, 2, 3) are the spatial parameterization angles of the depolarized subpolarizer;

[0079]

[0080] In the formula,

[0081]

[0082] In the formula, U4 is a unitary matrix, and U3 represents the depolarization transformation. Each set of two-dimensional factors in these submatrices of U3 has a geometric parameter regarding the latitude and longitude of the sphere, forming a spherical part for each plane. Therefore, each point in the unit cube has a cluster of three spheres, which can reflect the degrees of freedom in a specific direction and can be used to synthesize spectral lines with the same eigenvalues.

[0083] Furthermore, for the scattering medium, its depolarization is reflected at the main diagonal position, and at the same time, the symmetry completely eliminates the sphere, so the coherence matrix in Euclidean space can be obtained.

[0084]

[0085]

[0086] Here, U4(θ,δ) constitutes the unitary matrix of the eigenvector λ1 of the coherence matrix. There is no depolarization factor within U4(θ,δ), and the depolarization of the coherence matrix is ​​only related to λ. i (i = 2, 3, 4) are related. Because λ i (i = 2, 3, 4) is related to the depolarization matrix.

[0087]

[0088] μ 4i-4+j The results were obtained from four different sets of Boury matrices;

[0089]

[0090] This represents the transpose / conjugate operation, where i represents the imaginary number in the matrix; μ 4i-4+j (i,j=1,2,3,4) consists of four Pauli matrices σ i The results were obtained by combining (i = 1, 2, 3, 4).

[0091] The reflection coherence matrix and the depolarization matrix can be separated from the formula, and can be obtained through the inverse H→M transform.

[0092] in,

[0093]

[0094] The above formula can be transformed into

[0095]

[0096] In the formula, Δ=cos -1 (tan -1 2θ).

[0097] The proposed method for calculating elliptic parameters based on Mueller matrix dimensionality reduction effectively solves the technical challenge of complex calculations involving numerous Mueller matrix parameters in rough media. This invention enables rapid calculation of elliptic parameters and noise suppression, and features strong practicality, wide applicability, and high accuracy.

[0098] The present invention will now be described in detail with reference to the embodiments.

[0099] Example

[0100] This embodiment presents a method for calculating elliptic deviation parameters based on dimensionality reduction using the Mueller matrix, such as... Figure 1 As shown, the steps are as follows:

[0101] Step 1: Test the Mueller matrix of the rough surface material.

[0102] In step 1, preferably, a Mueller polarization imager is used as the test system.

[0103] The expression for the Mueller matrix is ​​as follows:

[0104] S' = MS

[0105] Where S' is the Stokes vector of the outgoing light, S(ε,θ) is the Stokes vector of the incident light, and M is the Mueller matrix of the rough surface material.

[0106] Step 2: Calculate the eigenvalues ​​of the coherence matrix corresponding to the material Mueller matrix and analyze its physical realizability, while performing noise reduction processing.

[0107] Step 3: Perform parameter optimization. The unitary matrix of the polarization part consists of 7 parameters α, β, γ, δ. i (i=1,2,3,4) is reduced to 2 parameters ψ,Δ, and the depolarization part is represented by only 3 eigenvalues.

[0108] Step 4: Decompose the Mueller matrix, establish the decomposed physical equations, calculate the ellipticity parameters, and analyze the experimental noise.

[0109] Error and accuracy analysis is performed on the results to determine the applicability and accuracy of the method.

[0110] The schematic diagram of the experimental apparatus used in this embodiment is shown below. Figure 2 As shown, the laser is a fiber laser with a wavelength of 632.8nm, L1, L2, and L3 are quartz quarter-wave plates, and P1, P2, and P3 are linear polarizers.

[0111] The detector is a CCD camera (image resolution: 800×600 pixels, 4096 gray levels). The Mueller matrix in this invention is obtained based on traditional Mueller matrix measurement methods, using polarization state generators (PSG, P2, and L2) and polarization state analyzers (PSA, P3, and L3). The incident light on the sample is measured using six incident Stokes vectors: linear horizontal, vertical, 45°, -45°, right-circularly polarized, and left-circularly polarized light, to generate polarized light with different polarization states. Thirty-six images are required to calculate the Mueller matrix of the sample. The experiment is conducted in an optically dark chamber.

[0112] like Figure 3 As shown, a green thin film sample was selected for simulation experiments. The sample was a green camouflage material with a surface roughness of approximately 0.9-1.2. Due to its rough surface, this sample could not be directly measured on an ellipsometer. The ellipsometric parameters calculated using this invention are as follows: Figure 4 As shown. In Figure 4 In this context, two parameters represent the relationship between the sample's polarization and the incident angle. As the incident angle increases, When Δ increases first and then decreases, the trend of θ changes more sensitively.

[0113] Figure 5 The comparison results of the LSM method and the Cloude method are shown in the figure. It can be found that the residual of the LSM method is smaller than that of the Cloude method. When the incident angle is 60°, the residual of the LSM method is about half of that of the Cloude method, and its decomposition accuracy is improved by about 50%.

[0114] The embodiments are merely illustrative of the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solution based on the technical concept proposed in this invention shall fall within the scope of protection of this invention.

Claims

1. A method for calculating ellipsometric parameters based on Mueller matrix dimension reduction, characterized in that, The method comprises the following steps: Testing the Mueller matrix of a scattering substance; Calculating eigenvalues of a coherence matrix corresponding to the Mueller matrix of the scattering substance and analyzing the eigenvalues; Constructing a reduced-dimension polarization space and a depolarization space of the scattering substance; Optimizing decomposition parameters, establishing a physical equation of decomposition, and performing ellipsometric parameter calculation and noise analysis by using an optimization method; Transforming a 4x4 complex space into a Euclidean space by using a reduced-dimension method: In the formula, U 4R represents a complex vector unit matrix, U3 is a 3x3 submatrix; t represents the transpose operation; U3 includes 3 submatrices, contains 6 parameters, in the scattering-induced depolarization process, U3 does not consider the eigenvector, only considers the eigenvalue thereof; then U4 includes 1 eigenvalue λ1, 2 eigenvector parameters (a, δ), the eigenvector corresponding to the eigenvalue λ1 includes 2 parameters, corresponding to a diagonal matrix, the diagonal matrix is in the form of In the formula, a and δ represent angle parameters of a polarization part in scattering, and U4 is a unitary matrix. U3 includes only 3 eigenvalues λ i and 0 eigenvectors, i = 2, 3, 4, so the coherence matrix representation of the target is The Mueller matrix in the Euclidean space is decomposed as M = (1 - d)M J + M D In the formula, d denotes the degree of depolarization coefficient, M D denotes the depolarization matrix related to scattering: Ellipsometric parameters are calculated by using a least square method: In the formula, ||.|| represents a 2-norm.

2. The method of claim 1, wherein, Polarization space with ellipsometric parameters denotes, denotes the amplitude ratio of s, p light components in the outgoing light, and Δ denotes the phase difference of s, p light; where r s , r p respectively represent the total reflection coefficients of s, p light, p s , p p represent the amplitudes of s, p light, and φ s φ p respectively represent the phases of s, p light.

3. The method of claim 1, wherein the Mueller matrix-based dimension reduction ellipsometric parameter calculation method is characterized by, A Mueller polarization imager is used as a test system, and an expression of the Mueller matrix is as follows: S' = MS In the formula, S' is a Stokes vector of outgoing light, S is a Stokes vector of incident light, and M is a Mueller matrix of a rough surface substance.

4. The method of claim 1, wherein, In step 2, a formula for calculating a coherence matrix of a scattering matrix is as follows: where M ij are Mueller matrix elements, σ i ,σ j are Poincare matrices.

5. The method of claim 1, wherein the Mueller matrix-based dimension reduction ellipsometric parameter calculation method is characterized by, Analysis of the eigenvalues includes physical realizability analysis, and it is judged whether the eigenvalues of the coherence matrix are greater than or equal to 0. If there is an eigenvalue less than 0 or a specific threshold value, it is indicated that the measurement of the Mueller matrix has noise and error, the eigenvalue needs to be set to 0, and normalization of the eigenvalues and reconstruction of the Mueller matrix are performed, and a calculation formula is as follows: In the formula, λ i denotes the eigenvalues of the coherence matrix, i = 1, 2, 3, tr denotes the trace of the matrix, M i denotes the sub-matrix of the Mueller matrix decomposition.

6. A computer device comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that, The processor implements the steps of the method of any one of claims 1-5 when executing the program.

7. A computer readable storage medium having stored thereon a computer program, characterized in that, The program is executed by the processor to implement the steps of the method of any one of claims 1-5.

8. A computer program product comprising a computer program, characterized in that, The computer program is executed by the processor to implement the steps of the method of any one of claims 1-5.

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