A model-free ellipsometer system parameter universal calibration method

Abstract

The application belongs to the technical field of precision optical instruments, and specifically discloses a general calibration method for system parameters of an ellipsometer not based on a model, which comprises the following steps: selecting standard samples of multiple categories as reference samples, placing the reference samples in an ellipsometer for polarization experiments, and collecting light intensity signals corresponding to the reference samples; obtaining light intensity matrices or light intensity coefficient matrices corresponding to the reference samples according to the light intensity signals of the reference samples; solving an analysis matrix describing an analyzing arm of the ellipsometer system and a modulation matrix describing a modulating arm of the ellipsometer system through multi-parameter joint optimization, and simultaneously obtaining characteristic parameters of the reference samples by using a nonlinear regression optimization method; applying the calibrated analysis matrix and the modulation matrix to a to-be-tested sample, and calculating based on a light intensity matrix or a light intensity coefficient matrix of the to-be-tested sample to solve optical parameters of the to-be-tested sample. The application can realize the generality and simplicity of the ellipsometer calibration process.

Description

Technical Field

[0001] This application belongs to the field of precision optical instrument technology, and more specifically, relates to a model-free universal calibration method for ellipsomer system parameters. Background Technology

[0002] An ellipsometer is an optical measuring instrument that uses the polarization properties of light to obtain information about a sample. Accurate measurement with an ellipsometer depends on the accurate calibration of its system parameters. Generally, the quantities to be calibrated mainly include key parameters such as the azimuth configuration of the polarization elements and the phase delay. Existing ellipsometer calibration methods can be divided into parameter calibration methods based on system models and calibration methods not based on system models. Model-free calibration methods are currently only applicable to full Mueller matrix ellipsometers and are also called eigenvalue calibration methods. Parameter calibration methods require establishing a rigorous and accurate optical system model and calibrating the parameters to be calibrated in the system by measuring common standard samples. Eigenvalue calibration methods consider the polarizing arm and analyzer arm of the full Mueller matrix ellipsometer as a whole. By measuring a specific standard sample and calculating the Mueller matrix of the reference sample, the modulation matrix of the polarizing arm, and the analysis matrix of the analyzer arm through matrix eigenvalue calculation, the calibration is completed.

[0003] Compared with parameter calibration, eigenvalue calibration eliminates the need for complex physical model derivation, does not require decoupling between parameters to be calibrated, and does not require precise reference sample information. It can effectively simplify the system calibration process, but it can only be applied to full Mueller matrix ellipsometers. For some Mueller matrix ellipsometers, there is currently no model-free calibration method. The system model derivation process is very cumbersome during calibration, and some systems need to ignore some system parameters or make approximations to simplify the calibration process.

[0004] In summary, how to achieve universality and simplicity in the ellipsometer calibration process is a technical problem that urgently needs to be solved. Summary of the Invention

[0005] In view of the shortcomings of the prior art, the purpose of this application is to provide a model-free universal calibration method for ellipsometer system parameters, aiming to achieve universality and simplicity in the ellipsometer calibration process.

[0006] To achieve the above objectives, in a first aspect, this application provides a model-free, universal calibration method for ellipsometry system parameters, characterized by comprising: Multiple categories of standard samples are selected as reference samples. The reference samples are placed in an elliptic polarimeter to conduct a polarization experiment, and the light intensity signal corresponding to each reference sample is collected. The light intensity matrix or light intensity coefficient matrix corresponding to each reference sample is obtained based on the light intensity signal of each reference sample; the light intensity matrix or light intensity coefficient matrix includes system parameters and sample information; Based on the light intensity matrix or light intensity coefficient matrix, the analysis matrix describing the polarizer arm and the modulation matrix describing the polarizer arm of the elliptic polarizer system are solved by multi-parameter joint optimization, and the characteristic parameters of the reference sample are obtained simultaneously by nonlinear regression optimization method. The calibrated analysis matrix and modulation matrix are applied to the sample under test, and calculations are performed based on the light intensity matrix or light intensity coefficient matrix of the sample under test to solve for the optical parameters of the sample under test.

[0007] Optionally, the light intensity matrix or light intensity coefficient matrix is ​​as shown in the following formula:

[0008] Where D is the light intensity matrix or light intensity coefficient matrix, W is the modulation matrix of the polarizer arm, and A is the analysis matrix of the analyzer arm. The Mueller matrix of the reference sample; The relationship between the Mueller matrix and the light intensity signal is shown in the following formula:

[0009] in, It is a light intensity signal. and These are the starting arm vector and the analyzing arm vector, respectively.

[0010] Optionally, the modulation type of the elliptic polarizer includes discrete modulation and continuous modulation. The method for obtaining the discrete modulation type light intensity matrix or light intensity coefficient matrix includes: By changing the polarization state of the polarizing arm and the polarizing analyzer arm respectively, the light intensity under each combination of polarization states is discretely obtained, and the obtained light intensities are arranged and combined to obtain the light intensity matrix. The method for obtaining the continuous modulation type light intensity matrix or light intensity coefficient matrix includes: Acquire a continuous light intensity signal that modulates over time, and determine the polarizer base vector corresponding to the polarizer modulation principle based on the polarizer arm vector and modulation matrix. Based on the polarizer arm vector and analysis matrix, the polarizer arm basis vector corresponding to the polarizer arm modulation principle is determined. ; A calculation model is established for the continuous light intensity signal with respect to the polarizer arm basis vector, the light intensity coefficient matrix, and the polarizer arm basis vector; The light intensity projection is performed using a nonlinear regression method or a Fourier transform method to obtain the light intensity coefficient matrix.

[0011] Optionally, the categories of the standard samples include air, linear polarizers, linear waveplates, and uniform isotropic thin film samples; Among them, the Mueller matrix corresponding to air is the identity matrix; The Mueller matrices of the linear polarizer, linear waveplate, and uniform isotropic thin film sample are shown below:

[0012] in, These are all characteristic parameters of the reference sample. It refers to the azimuth angle. The azimuth angle of linear polarizers and linear waveplates is adjustable, while the azimuth angle of isotropic thin films is not adjustable. It is a 4×4 rotation matrix; the isotropic thin film is used for oblique incidence configuration, and air, linear polarizer and linear waveplate are used for straight-through configuration.

[0013] Optionally, the process of solving the Mueller matrix of the linear polarizer and the linear waveplate includes: The azimuth angle is changed by rotating or flipping the linear polarizer and linear waveplate at equal intervals, and the azimuth angle relationship is as follows:

[0014]

[0015] in, and These are the azimuth angles before and after the linear polarizer and linear waveplate are flipped, respectively. It is the first of the equally spaced rotating linear polarizers and linear waveplates. The azimuth angle corresponding to the next measurement. It is the initial azimuth angle. They are equal azimuth intervals.

[0016] Optionally, the multi-parameter joint optimization constructs an objective function based on the least squares method or maximum likelihood estimation, and the objective function is shown in the following formula:

[0017] in, This refers to the number of reference samples. It is the first The light intensity matrix or light intensity coefficient matrix corresponding to each reference sample It is the first Muller matrices of one reference sample; During the solution process, the modulation matrix, analysis matrix, and characteristic parameters of each reference sample are optimized simultaneously, and the azimuth constraints of the reference samples are added to ensure the uniqueness of the system matrix solution.

[0018] Optionally, the calculation method for the optical parameters of the sample to be tested includes matrix calculation or parameter optimization calculation; the matrix calculation is applicable to full Mueller matrix ellipsometers, and the parameter optimization calculation is applicable to full Mueller matrix ellipsometers and partial Mueller matrix ellipsometers. The matrix calculation process is shown in the following formula:

[0019] in, It is the Mueller matrix of the sample to be tested. and These are the normalized analysis matrix and modulation matrix, respectively; The parameter optimization process is shown in the following formula:

[0020] in, For the optimized parameters of the sample to be tested, These are the parameters to be determined for the sample to be tested. It is the Mueller matrix corresponding to the parameters to be determined.

[0021] Optionally, before the calculation of the optical parameters of the sample to be tested, the following steps are also included: Normalize the first row and first column elements of the calibrated modulation matrix and analysis matrix to obtain the normalized modulation matrix and analysis matrix; Following the calculation of the optical parameters of the sample under test, the process also includes: The first row and first column elements of the optical parameters of the sample to be tested are normalized.

[0022] In a second aspect, this application provides an electronic device, comprising: at least one memory for storing a program; and at least one processor for executing the program stored in the memory, wherein when the program stored in the memory is executed, the processor is configured to execute the method described in the first aspect or any possible implementation thereof.

[0023] Thirdly, this application provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to perform the method described in the first aspect or any possible implementation thereof.

[0024] Fourthly, this application provides a computer program product that, when run on a processor, causes the processor to perform the method described in the first aspect or any possible implementation thereof.

[0025] It is understood that the beneficial effects of the second to fifth aspects mentioned above can be found in the relevant descriptions in the first aspect mentioned above, and will not be repeated here.

[0026] Overall, the technical solutions conceived in this application have the following beneficial effects compared with the prior art: (1) This application retains the advantages of the eigenvalue calibration method. It does not require complex system model derivation. The polarization arm and the polarization detection arm are regarded as a whole and represented by the modulation matrix and the analysis matrix respectively. The system calibration is transformed into solving the system matrix. The calibration can be completed by measuring a series of reference samples, which improves the universality of the calibration process. The operation process is simple and also ensures high calibration accuracy.

[0027] (2) This application does not require consideration of the coupling between parameters in the modulation matrix and the analysis matrix, and treats them as a whole to realize system calibration, which reduces the difficulty of system calibration. As a model-free method, it can calibrate the parameters to be calibrated that are ignored or approximated in model-based methods to simplify the model, such as the elliptic phase delay that may exist in the waveplate or the optical rotation that may exist in the polarization element, making the calibration more accurate.

[0028] (3) This application measures a series of calibration samples without needing to refer to the sample information. It also considers the reciprocity of the Mueller matrices of linear polarizers and linear waveplates and the characteristics of equal azimuth rotation, adding new constraints during the solution process to ensure the uniqueness and stability of the system matrix solution. At the same time, the system matrix obtained from the calibration can be directly used for the measurement of the sample to be tested.

[0029] (4) Unlike the eigenvalue calibration method, this application is only applicable to full Mueller matrix ellipsometers and is applicable to ellipsometers with various modulation principles. It is also applicable to full Mueller matrix and partial Mueller matrix ellipsometers, such as discrete modulation type, double rotation type, double photoelastic and four photoelastic modulation type, etc., and is more versatile. Attached Figure Description

[0030] Figure 1 This is a schematic diagram of the optical path of the transmission system of the elliptic polarizer in the embodiments of this application; Figure 2 This is a schematic diagram of the optical path of the reflective elliptic polarizer system in an embodiment of this application; Figure 3 This is a calibration flowchart summarizing the embodiments of this application; Figure 4 This is a complete calibration flowchart of the method in the embodiments of this application; Figure 5 This is a comparison diagram of the azimuth angles before and after the linear polarizer and linear waveplate are flipped; Figure 6This is a diagram showing the azimuth relationship between equally spaced linear polarizers and linear waveplates. Figure 7 This is a comparison diagram between the Muller matrix elements of the multi-azimuth polarizer and waveplate and the reference value, measured by the calibration method proposed in this application, with the configuration in Table 1 as an example, and 27dB noise added to the detected light intensity, using the configuration method proposed in this application. Figure 8 This is a schematic diagram of the structure of an electronic device according to an embodiment of this application.

[0031] In all the accompanying drawings, the same reference numerals are used to denote the same elements or structures, wherein: 101 is the light source, 102 is the collimating lens, 103 is the polarizer, 104 is the polarizer arm modulation unit, 105 is the sample position, 106 is the polarizer arm modulation unit, 107 is the polarizer, 108 is the collecting lens, and 109 is the detector. Detailed Implementation

[0032] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0033] In this article, the term "and / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, or B existing alone. The symbol " / " in this article indicates that the related objects are in an "or" relationship; for example, A / B means A or B.

[0034] The terms "first" and "second," etc., used in the specification and claims herein are used to distinguish different objects, not to describe a specific order of objects. For example, "first response message" and "second response message," etc., are used to distinguish different response messages, not to describe a specific order of response messages.

[0035] In the embodiments of this application, the terms "exemplary" or "for example" are used to indicate that something is an example, illustration, or description. Any embodiment or design that is described as "exemplary" or "for example" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or design. Specifically, the use of the terms "exemplary" or "for example" is intended to present the relevant concepts in a specific manner.

[0036] In the description of the embodiments of this application, unless otherwise stated, "multiple" means two or more, for example, multiple processing units means two or more processing units, multiple elements means two or more elements, etc.

[0037] The embodiments of this application are described below with reference to the accompanying drawings.

[0038] Please see Figure 1 and Figure 2 Ellipsometry, also known as an ellipsometer, specifically includes direct-through and reflective types. The optical path diagram of a direct-through ellipsometer system is shown below. Figure 1 As shown, the instrument is configured as follows: light source 101, collimating lens 102, polarizer 103, polarizer arm modulation unit 104, sample position 105, analyzer arm modulation unit 106, analyzer 107, collecting lens 108, and detector 109. The light source 101 and collimating lens 102 form the light source module, providing a parallel illumination beam to the system; the polarizer 103 and polarizer arm modulation unit 104 form the polarizer arm, used to modulate the polarization state of the beam incident on the sample; the analyzer arm modulation unit 106 and analyzer 107 form the analyzer arm, used to analyze the polarization state of the light emitted after passing through the sample. The collecting lens and detector collect the light intensity signal.

[0039] The optical path diagram of an ellipsometer oblique incidence system, i.e., a reflective system, is shown below. Figure 2 As shown, the instrument is configured as follows: light source 101, collimating lens 102, polarizer 103, polarizer arm modulation unit 104, sample position 105, analyzer arm modulation unit 106, analyzer 107, collecting lens 108, and detector 109. It should be noted that this method is applicable to ellipsometers with various modulation principles, including discrete modulation types (such as liquid crystal modulation, rotation compensators, or polarizers to several specific azimuth angles) and continuous modulation types (such as continuous rotation compensators or photoelastic modulation). The polarizer arm modulation unit 104 and analyzer arm modulation unit 106 differ depending on the modulation principle; for example, liquid crystal modulation corresponds to a liquid crystal modulator, rotation compensators correspond to waveplates, and photoelastic modulation corresponds to a photoelastic modulator (PEM). This method is also applicable to the system calibration of partial and full Mueller matrix ellipsometers. The number of parameters that an ellipsometer can measure depends on the number of independent polarization state combinations of the polarizing arm and the analyzer arm. Therefore, it is limited by the number of modulators in the polarizing arm modulation unit 104 and the analyzer arm modulation unit 106. For example, a dual photoellipse can only measure 9 parameters, so it can only measure a partial Mueller matrix. A quad photoellipse can measure more than 16 parameters, so it can be used to measure the full Mueller matrix. A dual liquid crystal can only measure partial Mueller matrix elements, while a quad liquid crystal can measure the full Mueller matrix elements.

[0040] The calibration of the oblique incidence system is based on the calibration of the through-pass system. Calibration is performed by measuring an isotropic thin film sample to determine the system's incident angle. Taking the transmission system as an example, the calibration principle is equally applicable to the reflection system. Similarly, this method can be used for multi-wavelength calibration. Calibration at different wavelengths is independent and uses the same calibration approach; therefore, only the calibration steps for a single wavelength are described here. The specific calibration scheme is as follows... Figure 3 As shown.

[0041] S101. Select multiple categories of standard samples as reference samples, place the reference samples in an elliptic polarimeter to perform a polarization experiment, and collect the light intensity signal corresponding to each reference sample. S102. Obtain the light intensity matrix or light intensity coefficient matrix corresponding to each reference sample based on the light intensity signal of each reference sample; the light intensity matrix or light intensity coefficient matrix includes system parameters and sample information; S103. Based on the light intensity matrix or light intensity coefficient matrix, the analysis matrix describing the polarizer arm and the modulation matrix describing the polarizer arm of the elliptic polarizer system are solved by multi-parameter joint optimization, and the characteristic parameters of the reference sample are obtained simultaneously by using a nonlinear regression optimization method. S104. Apply the calibrated analysis matrix and modulation matrix to the sample under test, and calculate the optical parameters of the sample under test based on the light intensity matrix or light intensity coefficient matrix of the sample under test.

[0042] Specifically, in this embodiment, firstly, multiple standard samples of known categories are selected as reference samples in step S101, and then each reference sample is placed sequentially on the sample stage of the ellipsometer for polarization experiments. Through the coordinated modulation of the polarizing arm and the analyzing arm, the light intensity signal corresponding to each reference sample under different combinations of polarization states is acquired.

[0043] Secondly, according to the modulation type of the ellipsometer, S102 processes the acquired light intensity signals of each reference sample to obtain the corresponding light intensity matrix or light intensity coefficient matrix. For discrete modulation ellipsometers, the light intensity under each polarization state combination is discretely obtained by changing the polarization state of the polarizing arm and the analyzing arm, and these are arranged and combined to form the light intensity matrix. For continuous modulation ellipsometers, the light intensity signal is acquired by continuously modulating over time, and the light intensity is projected by introducing the basis vector corresponding to the modulation principle. The light intensity coefficient matrix is ​​obtained using methods such as nonlinear regression or Fourier transform. The light intensity matrix or light intensity coefficient matrix is ​​structured measurement data, which simultaneously contains the system parameters of the ellipsometer and the sample information of the reference samples, serving as input data for solving the system matrix and sample parameters.

[0044] Through S103, a multi-parameter joint optimization objective function is constructed based on the light intensity matrix or light intensity coefficient matrix corresponding to all reference samples. This optimization process uses least squares or maximum likelihood estimation as criteria, aiming to minimize the sum of squared residuals between the measured and theoretical values ​​of the light intensity matrix or light intensity coefficient matrix. Simultaneously, it solves for three unknowns: the analysis matrix describing the overall characteristics of the elliptic polarizer's analyzer arm, the modulation matrix describing the overall characteristics of the polarizer arm, and the characteristic parameters of each reference sample (such as transmittance, retardation, azimuth angle, etc.). During the solution process, the uniqueness of the system matrix solution and the stability of the solution process are ensured by introducing azimuth constraints on the reference samples (such as equidistant rotation or flipping relationships). The core of this step lies in transforming the system calibration problem into a system matrix solution problem, without relying on complex system physical model derivations.

[0045] Finally, the analysis matrix and modulation matrix obtained in step S103 are used as the calibrated system matrix and applied to the measurement of the sample under test. The same polarization experiment as the reference sample is performed on the sample under test, its light intensity signal is collected, and the corresponding light intensity matrix or light intensity coefficient matrix is ​​obtained according to the method in step S102.

[0046] Based on this, the optical parameters of the sample under test are solved using either matrix calculation or parameter optimization. For a full Mueller matrix ellipsometer, the Mueller matrix of the sample can be directly calculated by taking the generalized inverse of the matrix. For both full and partial Mueller matrix ellipsometers, parameter optimization is used to find the sample parameters that minimize the residual between the measured light intensity matrix and the theoretical light intensity matrix, which are then used as the final measurement result. Before and after the calculation, the first row and first column elements of the system matrix and the optical parameters of the sample under test need to be normalized to eliminate the influence of the absolute amplitude of light intensity and obtain a physically meaningful relative measurement result.

[0047] This embodiment provides a model-free, general system calibration method for ellipsometers, referring to... Figure 4 The system calibration scheme includes: Step 1: Set up the experimental optical path of the ellipsometer as follows Figure 1 or Figure 2 As shown, its system model can be described as follows: S in and S out M represents the Stokes vectors of the incident and emitted light intensities, respectively. PSG and M PSA The Muller matrices describing the polarization modulation unit and the polarization modulation unit, respectively, are M. P and M A The Mueller matrices for the polarizer and analyzer are described separately. Since the detector can only sense light intensity information, the above equation can be further expressed as: The parameters to be calibrated on the polarization arm and the polarization detector arm generally refer to the azimuth configuration of the polarization element and key modulation parameters, such as phase delay, throughout the measurement process.

[0048] Step 2: Prepare reference samples for polarization experiments. For the direct-through mode, select a series of transmissive standard samples, such as air, linear polarizers, and linear waveplates. For the oblique-incident mode, select reflective standard samples, such as isotropic thin films, based on the above transmissive standard samples. To ensure accurate system calibration, air, linear polarizers at several angles, and linear waveplates are generally selected in the direct-through mode, while an isotropic thin film is added in the oblique-incident mode. For standard samples such as linear polarizers and linear waveplates, a flipped or equally spaced rotation mounting method can be chosen. Utilizing the reciprocity of their Mueller matrix and its azimuth-dependent characteristics, more effective information is introduced without introducing more parameters to be calibrated, effectively improving the uniqueness and robustness of solving the system matrix. It should be noted that the above is not the only option, but only a feasible combination of reference samples. Theoretically, the more samples or azimuth angles selected, the better the robustness of the system calibration.

[0049] Step 3: Place the selected reference samples sequentially on sample position 105, ensuring that the linear polarizer and linear waveplate are perpendicular to the system optical path, and the isotropic thin film is perpendicular to the incident plane of the optical path. Based on the ellipsometer system, perform polarization experiments on the reference samples sequentially and collect experimental light intensity signals.

[0050] Step 4: Obtain the light intensity matrix or light intensity coefficient matrix containing system parameters and sample information from the light intensity signal. The ellipsometer includes discrete modulation and continuous modulation types. It should be noted that the methods for obtaining the discrete modulation type light intensity matrix or light intensity coefficient matrix include: By changing the polarization state of the polarizing arm and the polarizing analyzer arm respectively, the light intensity under each combination of polarization states is discretely obtained, and the obtained light intensities are arranged and combined to obtain the light intensity matrix. The method for obtaining the continuous modulation type light intensity matrix or light intensity coefficient matrix includes: Acquire a continuous light intensity signal that modulates over time, and determine the polarizer base vector corresponding to the polarizer modulation principle based on the polarizer arm vector and modulation matrix. Based on the polarizer arm vector and analysis matrix, the polarizer arm basis vector corresponding to the polarizer arm modulation principle is determined. ; A calculation model is established for the continuous light intensity signal with respect to the polarizer arm basis vector, the light intensity coefficient matrix, and the polarizer arm basis vector; The light intensity projection is performed using a nonlinear regression method or a Fourier transform method to obtain the light intensity coefficient matrix.

[0051] Specifically, the discrete modulation ellipsometer obtains the light intensity of each polarization state combination by separately changing the polarization state of the polarizing arm and the polarization state of the ellipsoid arm, and then arranges and combines them to obtain the light intensity matrix D, as shown below:

[0052] Where A and W are the analysis matrix of the analyzer arm and the modulation matrix of the polarizer arm, respectively, i.e., the system matrix; the continuously modulated ellipsometer generally acquires a series of light intensities that are continuously modulated over time. By introducing the analyzer arm basis vector and the polarizer arm basis vector, which contain the dynamic parameters of the polarization system, a calculation model of the light intensity signal with respect to the analyzer arm basis vector, the light intensity projection matrix, and the polarizer arm basis vector is established. The light intensity projection matrix, i.e., the light intensity coefficient matrix, is obtained using a nonlinear regression method.

[0053] in, It is the transpose of the basis vector corresponding to the starting arm. This is the basis vector corresponding to the polarization analyzer. The design of the basis vector needs to be based on the modulation principle and modulation unit composition of the polarization analyzer and polarization arm.

[0054] Step 5: After obtaining a series of light intensity matrices or light intensity coefficient matrices for the selected reference sample, the system matrix and the parameters of the reference sample are solved through multi-parameter optimization. The Mueller matrix for air is considered a 4×4 identity matrix, and the Mueller matrices for other standard samples are modeled as follows:

[0055] in τ , a , b , c , θ All of these are quantities to be calibrated, i.e., characteristic parameters of the reference sample. θ R(·) is its azimuth angle, and R(·) is a 4×4 rotation matrix. Isotropic films are used for oblique incidence configurations, while air, linear polarizers, and linear waveplates are used for straight-through configurations. The azimuth angles of linear polarizers and linear waveplates are adjustable, corresponding to their fast-axis azimuth angles. The azimuth angle of the isotropic film corresponds to the angle between the incident plane of light and the plane perpendicular to the sample surface, and is generally not adjustable. For example... Figure 5 and Figure 6 As shown, the azimuth relationships between the reciprocity of the linear polarizer and the linear waveplate and the equidistant selection are as follows:

[0056]

[0057] in, and These are the azimuth angles before and after the linear polarizer and linear waveplate are flipped, respectively. It is the first of the equally spaced rotating linear polarizers and linear waveplates. The azimuth angle corresponding to the next measurement. It is the initial azimuth angle. They are equal azimuth intervals.

[0058] When solving for the system matrix using a combination of reference samples, an objective function is derived based on the ideas of least squares or maximum likelihood estimation, and then solved using a multi-objective optimization method. Taking least squares as an example, the objective function is as follows:

[0059] N This represents the number of selected samples, where A and W are the parameters of the reference sample. τ , a , b , c , θ All are values ​​to be optimized. A is an m×4 matrix, W is a 4×n matrix, and D is an m×n matrix. For the discrete modulation ellipsometer, n is determined by the number of polarization states of the polarizing arm, and m is determined by the number of polarization states of the analyzing arm. For the continuous modulation ellipsometer, n and m are determined by the number of basis functions in the basis vectors of the polarizing arm and analyzing arm, respectively. It should be noted that for reference samples with different azimuth angles, the parameters of the same reference sample ( τ , a , b , c To maintain consistency, the azimuth information, combined with the installation method of the linear polarizers and linear waveplates, should be accompanied by the addition of reciprocity constraints or equal spacing constraints. Generally, for a full Mueller matrix ellipsometer, the selected reference sample can be consistent with the eigenvalue calibration method, including air, two azimuth linear polarizers, and one azimuth linear waveplate. For partial Mueller matrix ellipsometers, the selected reference sample for a full Mueller matrix ellipsometer is sufficient for solving the system matrix and reference sample parameters. Increasing the number of reference samples and adding azimuth constraints can effectively improve the uniqueness and robustness of the solution.

[0060] Step 6: The system matrix obtained above can be directly used for the measurement of the sample to be tested. Before directly using it for the measurement of the sample to be tested, the elements in the first row and first column of the system matrix A and W need to be normalized, and the elements in the first row and first column of the Mueller matrix of the sample to be tested also need to be normalized. The specific measures are as follows: place the sample to be tested on the sample stage of the ellipsometer, obtain the light intensity signal according to the established procedure of the ellipsometer sample measurement, obtain the light intensity matrix or light intensity coefficient matrix D according to the steps in step 4, and solve for the Mueller matrix or the parameters to be measured of the sample to be tested through matrix calculation or parameter optimization. The specific formula is as follows:

[0061]

[0062] in, To obtain the Mueller matrix of the sample to be tested, the matrix calculation method is only applicable to full Mueller matrix ellipsometers. P These are the parameters to be determined for the sample to be tested. and The obtained system matrices A and W are normalized according to the elements in the first row and first column. This method is applicable to the direct solution of the sample under test in both full Mueller matrix ellipsometers and partial Mueller matrix ellipsometers. It should be noted that for partial Mueller matrix ellipsometers, the parameters to be determined... P The sample information parameters cannot exceed those that the Mueller matrix ellipsometer itself can measure.

[0063] In one embodiment, the ellipsometer is a direct-through four-photoelastic full Mueller matrix ellipsometer, which belongs to the continuous modulation type ellipsometer. The light source 101 is a parallel laser tube with an emission wavelength of 633nm and a noise level of 27dB. The polarizer 103 and the analyzer 107 are conventional commercial linear polarizers. The polarizer arm modulation unit 104 and the analyzer arm modulation unit 106 are both two photoelastic modulators with different modulation frequencies. The two photoelastic modulators of the polarizer arm modulation unit are the first photoelastic modulator and the second photoelastic modulator, respectively. The two photoelastic modulators of the analyzer arm modulation unit are the third photoelastic modulator and the fourth photoelastic modulator, respectively. The detector 109 is a photomultiplier tube adapted to the wavelength range of the light source and is equipped with a suitable high-speed data acquisition card.

[0064] The specific calibration and measurement process is as follows: In the first step, the actual optical path plane is used as the reference 0° azimuth angle. In this embodiment, the azimuth angles are all relative to the reference 0° azimuth angle. The azimuth angles of each device obtained by calibration in the following text will not be described as relative azimuth angles. A through-type four-optical-elastic full Mueller matrix ellipsometer is built according to the system configuration shown in Table 1.

[0065] Table 1 System Configuration Table

[0066] For a single measurement sample, the detector acquires the measurement light intensity signal. The theoretical calculation model is as follows:

[0067]

[0068] (1) in, S in M represents the Stokes vector of the light emitted from the light source. P M S M A These are the Mueller matrices for the polarizer, sample, and analyzer, respectively. The Mueller matrix represents a phase delay unit with a phase delay of δ. and These are the azimuth angles of the polarizer and the analyzer, respectively. and ( k =1,2,3,4) represent the first, second, third, and fourth numbers respectively. k The azimuth angle and phase delay of each photoelastic modulator, where the phase delay of the photoelastic modulator is: , where F, , , The parameters to be calibrated in this system are, in order: peak delay, frequency, initial phase, and static delay of the photoelastic modulator. When establishing the optical model of the system based on the Jones-Muller matrix principle, the parameters to be calibrated in this system include: (1) the azimuth angles of the polarizer, analyzer, and four photoelastic modulators; (2) the peak delay, modulation frequency, initial phase, and static delay of each of the four photoelastic modulators. According to the properties of the above system parameters, they are divided into dynamic system parameters and static system parameters. Dynamic system parameters refer to parameters that may change in each measurement, including the peak delay, modulation frequency, and initial phase of each photoelastic modulator. These parameters may fluctuate in each measurement due to environmental influences and unstable control, but they can all be obtained through the reference signal of the photoelastic modulator. Static system parameters refer to parameters that remain stable in each measurement, including the azimuth angle information of the polarization elements in the system and the static delay of each photoelastic modulator. They need to be precisely calibrated because each polarization element in the four-photoelastic modulated Muller matrix polarimeter is fixed, and the static delay of the photoelastic modulator is only related to the material and its geometry.

[0070] In the second step, the ellipsometer is a through-type full Mueller matrix ellipsometer. Transmission-type standard samples, including air, linear polarizers at several azimuth angles, and linear waveplates, are selected as reference samples. The initial azimuth angle of the linear polarizers is set to 0°, and three azimuth angles of linear polarizers and a linear quarter-waveplate are selected, with a nominal interval of 30°. As can be seen from the eigenvalue calibration method, air, two azimuth angles of polarizers, and one azimuth angle of waveplate are sufficient to determine a unique system matrix. Choosing more azimuth angles here ensures the uniqueness and robustness of the solved system matrix.

[0071] In the third step, the reference samples selected in the second step are placed sequentially on sample position 105. Polarization experiments are then performed on the reference samples sequentially using the four-photoelastic full Mueller matrix ellipsometer system, and experimental light intensity signals are collected. It is important to ensure a sufficient number of data points and to select an appropriate sampling frequency that satisfies the Nyquist sampling theorem to ensure that the collected signal is not distorted and to reduce the impact of random noise on the acquisition of the light intensity coefficient matrix.

[0072] In the fourth step, appropriate basis vectors were selected based on the number of photoelastic modulators on the polarizing arm and the analyzing arm, where the polarizing arm basis vector is: The basis vector of the bias arm is That is, based on Formula 1, for each time t, It is a 9×1 vector. Given a 1×9 vector, a 9×9 light intensity projection matrix D is obtained from the time-domain continuously modulated light intensity signal, as follows: (2) (3) (4) in, , represents the dynamic delay part of the phase delay of the photoelastic modulator, and the subscript indicates the label of the photoelastic modulator; the basis vector contains all the dynamic system parameters in the system, A and W are the analyzer arm analysis matrix and the polarizer arm modulation matrix, respectively, and D is the light intensity projection matrix. A is a 9×4 matrix, W is a 4×9 matrix, and D is a 9×9 matrix. The polarizer arm modulation matrix W contains the azimuth angle parameters of the polarization element in the polarizer arm and the static delay parameter of the photoelastic modulator; the analyzer arm analysis matrix A contains the azimuth angle parameters of the polarization element in the analyzer arm and the static delay parameter of the photoelastic modulator. The above formula (1) is the system model of the well-known four-photoelastic modulated Mueller matrix ellipsometer, which can be written in the form of formula (2) by introducing the basis vector.

[0073] In step five, the models for the linear polarizer and the linear waveplate are as follows: (5) (6) The parameters to be calibrated for the linear polarizer are as follows: P P =( τ 1, a 1, b 1, c 1, θ init1 , θ d1 The parameters to be calibrated for the linear waveplate are: P C =( τ 2, a 2, b 2, c 2, θ init2 , θ d2 The number of selected samples is 7. The objective function based on the least squares method is as follows. The system matrix A and W are solved by combining a suitable multi-parameter optimization algorithm.

[0074] (7) in, That is, the optimization result obtained by solving based on formula (7).

[0075] In step six, before directly using it for measurement of the sample to be tested, the system matrix needs to be... and The elements in the first row and first column are normalized. The selected test samples are linear polarizers and linear waveplates with an azimuth range of 0-180° and an azimuth interval of 10°. The specific measures are as follows: the test sample is placed on the sample stage of the four-photoelastic ellipsometer, and the light intensity signal is obtained according to the established procedure of the four-photoelastic ellipsometer for measuring the sample. The light intensity coefficient matrix D is obtained according to the fourth step. Since the four-photoelastic ellipsometer is a full Mueller matrix ellipsometer, the full Mueller matrix of the test sample can be solved by matrix calculation or parameter optimization. The specific formula is as follows: (8) (9) in, To obtain the Mueller matrix of the sample to be tested, the matrix calculation method is only applicable to the full Mueller matrix ellipsometer. The parameters to be determined for the sample are the full Mueller matrix. and The system matrix obtained in step 5 and The matrix is ​​obtained by normalizing the elements in the first row and first column. Finally, the elements in the first row and first column of the Mueller matrix of the sample to be tested are normalized. Figure 7In this example, the calibration method proposed in the application is used to solve the Mueller matrix elements of the test sample based on formula (8) and compare them with the reference values. The results show a good match, which proves the effectiveness and stability of this method.

[0076] Compared to existing model-based calibration methods for four-electroelastic ellipsometers, the parameter calibration method provided in this embodiment retains the advantages of eigenvalue calibration. It eliminates the need for complex system model derivation, treating the polarizer and analyzer as separate entities, represented by modulation and analysis matrices respectively. This transforms system calibration into solving the system matrix, which can be completed through measurements of a series of reference samples. The operation is simple and ensures high calibration accuracy. Furthermore, it eliminates the need to consider the coupling between parameters in the modulation and analysis matrices, treating them as a whole for system calibration, thus reducing the difficulty of system calibration. As a non-based... The model-based method can calibrate high-frequency components and azimuth errors that are ignored or approximated in order to simplify the model, making the calibration more accurate. Considering the equal azimuth rotation characteristics of the Mueller matrices of linear polarizers and linear waveplates, new constraints are added during the solution process to ensure the uniqueness and stability of the system matrix solution. By simply changing the basis vectors of the polarizer arm and the analyzer arm and the dimension of the system matrix, this method can be applied to the system calibration of the dual-optical-elastic partial Mueller matrix ellipsometer system, breaking through the limitation of the eigenvalue calibration method being only applicable to the full Mueller matrix ellipsometer, and demonstrating better versatility.

[0077] Reference Figure 8 Based on the methods in the above embodiments, this application provides an electronic device that may include: a processor 810, a communications interface 820, a memory 830, and a communication bus 840. The processor 810, communications interface 820, and memory 830 communicate with each other via the communication bus 840. The processor 810 can call logical instructions in the memory 830 to execute the methods in the above embodiments.

[0078] Furthermore, the logical instructions in the aforementioned memory 830 can be implemented as software functional units and, when sold or used as independent products, can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or a portion of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this application.

[0079] Based on the methods in the above embodiments, this application provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to execute the methods in the above embodiments.

[0080] Based on the methods in the above embodiments, this application provides a computer program product that, when run on a processor, causes the processor to execute the methods in the above embodiments.

[0081] It is understood that the processor in the embodiments of this application can be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, transistor logic devices, hardware components, or any combination thereof. A general-purpose processor can be a microprocessor or any conventional processor.

[0082] The method steps in this application embodiment can be implemented in hardware or by a processor executing software instructions. The software instructions can consist of corresponding software modules, which can be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, hard disks, portable hard disks, CD-ROMs, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor, enabling the processor to read information from and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can reside in an ASIC.

[0083] In the above embodiments, implementation can be achieved entirely or partially through software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented entirely or partially as a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted through the computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state disk (SSD)).

[0084] It is understood that the various numerical designations used in the embodiments of this application are merely for the convenience of description and are not intended to limit the scope of the embodiments of this application.

[0085] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of this application and is not intended to limit this application. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A model-free, universal calibration method for ellipsomerometer system parameters, characterized in that, include: Multiple categories of standard samples are selected as reference samples. The reference samples are placed in an elliptic polarimeter to conduct a polarization experiment, and the light intensity signal corresponding to each reference sample is collected. The light intensity matrix or light intensity coefficient matrix corresponding to each reference sample is obtained based on the light intensity signal of each reference sample; the light intensity matrix or light intensity coefficient matrix includes system parameters and sample information; Based on the light intensity matrix or light intensity coefficient matrix, the analysis matrix describing the polarizer arm and the modulation matrix describing the polarizer arm of the elliptic polarizer system are solved by multi-parameter joint optimization, and the characteristic parameters of the reference sample are obtained simultaneously by nonlinear regression optimization method. The calibrated analysis matrix and modulation matrix are applied to the sample under test, and calculations are performed based on the light intensity matrix or light intensity coefficient matrix of the sample under test to solve for the optical parameters of the sample under test.

2. The general calibration method for ellipsometry system parameters without model-based approach according to claim 1, characterized in that, The light intensity matrix or light intensity coefficient matrix is ​​shown in the following formula: Where D is the light intensity matrix or light intensity coefficient matrix, W is the modulation matrix of the polarizer arm, and A is the analysis matrix of the analyzer arm. The Mueller matrix of the reference sample; The relationship between the Mueller matrix and the light intensity signal is shown in the following formula: in, It is a light intensity signal. and These are the starting arm vector and the analyzing arm vector, respectively.

3. The general calibration method for ellipsometry system parameters without model-based approach according to claim 1, characterized in that, The modulation types of the elliptic polarizer include discrete modulation and continuous modulation. The method for obtaining the discrete modulation type light intensity matrix or light intensity coefficient matrix includes: By changing the polarization state of the polarizing arm and the polarizing analyzer arm respectively, the light intensity under each combination of polarization states is discretely obtained, and the obtained light intensities are arranged and combined to obtain the light intensity matrix. The method for obtaining the continuous modulation type light intensity matrix or light intensity coefficient matrix includes: Acquire a continuous light intensity signal that modulates over time, and determine the polarizer base vector corresponding to the polarizer modulation principle based on the polarizer arm vector and modulation matrix. Based on the polarizer arm vector and analysis matrix, the polarizer arm basis vector corresponding to the polarizer arm modulation principle is determined. ; A calculation model is established for the continuous light intensity signal with respect to the polarizer arm basis vector, the light intensity coefficient matrix, and the polarizer arm basis vector; The light intensity projection is performed using a nonlinear regression method or a Fourier transform method to obtain the light intensity coefficient matrix.

4. The general calibration method for ellipsometry system parameters without model-based approach according to claim 1, characterized in that, The categories of standard samples include air, linear polarizers, linear waveplates, and uniform isotropic thin film samples. Among them, the Mueller matrix corresponding to air is the identity matrix; The Mueller matrices of the linear polarizer, linear waveplate, and uniform isotropic thin film sample are shown below: in, These are all characteristic parameters of the reference sample. It refers to the azimuth angle. The azimuth angle of linear polarizers and linear waveplates is adjustable, while the azimuth angle of isotropic thin films is not adjustable. It is a 4×4 rotation matrix; the isotropic thin film is used for oblique incidence configuration, and air, linear polarizer and linear waveplate are used for straight-through configuration.

5. The general calibration method for ellipsometry system parameters without model-based approach according to claim 4, characterized in that, The process of solving the Mueller matrix for the linear polarizer and linear waveplate includes: The azimuth angle is changed by rotating or flipping the linear polarizer and linear waveplate at equal intervals, and the azimuth angle relationship is as follows: in, and These are the azimuth angles before and after the linear polarizer and linear waveplate are flipped, respectively. It is the first of the equally spaced rotating linear polarizers and linear waveplates. The azimuth angle corresponding to the next measurement. It is the initial azimuth angle. They are equal azimuth intervals.

6. The general calibration method for ellipsometry system parameters without model-based approach according to claim 1, characterized in that, The multi-parameter joint optimization is based on the least squares method or maximum likelihood estimation to construct the objective function, which is shown in the following formula: in, This refers to the number of reference samples. It is the first The light intensity matrix or light intensity coefficient matrix corresponding to each reference sample It is the first Muller matrices of one reference sample; During the solution process, the modulation matrix, analysis matrix, and characteristic parameters of each reference sample are optimized simultaneously, and the azimuth constraints of the reference samples are added to ensure the uniqueness of the system matrix solution.

7. The general calibration method for ellipsomerometer system parameters without model-based approach according to claim 1, characterized in that, The calculation methods for the optical parameters of the sample under test include matrix calculation or parameter optimization calculation; the matrix calculation is applicable to full Mueller matrix ellipsometers, and the parameter optimization calculation is applicable to full Mueller matrix ellipsometers and partial Mueller matrix ellipsometers. The matrix calculation process is shown in the following formula: in, It is the Mueller matrix of the sample to be tested. and These are the normalized analysis matrix and modulation matrix, respectively; The parameter optimization process is shown in the following formula: in, For the optimized parameters of the sample to be tested, These are the parameters to be determined for the sample to be tested. It is the Mueller matrix corresponding to the parameters to be determined.

8. The general calibration method for ellipsometry system parameters without model-based approach according to claim 1, characterized in that, Before the calculation of the optical parameters of the sample to be tested, the following steps are also included: Normalize the first row and first column elements of the calibrated modulation matrix and analysis matrix to obtain the normalized modulation matrix and analysis matrix; Following the calculation of the optical parameters of the sample under test, the process also includes: The first row and first column elements of the optical parameters of the sample to be tested are normalized.

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

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is run on the processor, it causes the processor to perform the method as described in any one of claims 1-8.

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