A Phase-Shift Interferometric In-Situ Measurement Method for Three-Frame Dynamic Mode Decomposition

Abstract

This invention provides an in-situ phase-shift interferometry measurement method based on three-frame dynamic mode decomposition, comprising: acquiring three-frame phase-shift interferograms of the object under test, and performing differential and normalization processing on the three-frame phase-shift interferograms to obtain a first interferogram sequence after removing the background; expanding the first interferogram sequence into column vector form and constructing a data matrix and a data matrix in chronological order; performing singular value decomposition on the data matrix to calculate an approximate matrix; performing feature analysis on the approximate matrix to extract eigenvalues ​​and eigenvectors, and mapping the eigenvectors back to the space of the first interferogram sequence to reconstruct the dynamic mode of the first interferogram sequence; and recovering the phase distribution of the object under test based on the argument information of the principal mode in the dynamic mode. This invention overcomes the limitation of traditional dynamic mode decomposition algorithms requiring at least four interferograms by constructing a new phase-shift interferogram sequence, and exhibits good robustness to background intensity fluctuations and noise interference.

Description

Technical Field

[0001] This invention relates to the field of optical interferometry technology, and in particular to a phase-shifting interferometric in-situ measurement method based on three-frame dynamic mode decomposition. Background Technology

[0002] Phase-shifting interferometry, due to its high measurement accuracy and non-contact detection capabilities, has become a fundamental technology in applications such as optical surface inspection, wavefront reconstruction, and temperature and stress analysis. In the semiconductor industry, as integrated circuit manufacturing evolves towards higher precision and smaller feature sizes, higher demands are placed on detection accuracy and efficiency. Optical interferometry, with its high resolution and non-contact detection characteristics, is widely used in semiconductor manufacturing processes for critical parameter detection, device defect identification, and online quality control, significantly improving production reliability and yield. The performance of phase-shifting interferometry largely depends on the phase-shifting algorithm employed. Typically, phase-shifting algorithms require multiple interferograms for phase demodulation, but their performance is easily affected by environmental disturbances such as turbulence, vibration, and phase shifter nonlinearity. In such cases, reducing the required number of frames helps improve measurement accuracy and real-time performance. Therefore, minimizing the required number of interferogram frames while ensuring phase demodulation accuracy has become a key focus of phase-shifting algorithm research.

[0003] In existing technologies, high-precision demodulation of the phase in interferograms can be achieved by extracting the principal modes and their eigenvalues, exhibiting good noise robustness. Since interferogram models typically contain three main modes, traditional dynamic mode decomposition (DynamD) techniques require at least four interferogram frames for phase demodulation. This requirement limits its application in highly dynamic or frame-limited conditions and places higher demands on data acquisition and processing. Attempts have combined spatial carrier phase shifting with Dynamic Mode Decomposition, constructing interferogram sequences from the spatial carrier in a single frame for phase demodulation, partially alleviating the multi-frame requirement of Dynamic Mode Decomposition. However, this does not fundamentally solve the limitation that Dynamic Mode Decomposition requires at least four interferogram frames.

[0004] Therefore, how to further minimize the required number of interferogram frames is a problem that needs to be solved by those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide a phase-shift interferometric in-situ measurement method for three-frame dynamic mode decomposition, which aims to solve or improve at least one of the above-mentioned technical problems.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] A phase-shift interferometric in-situ measurement method for three-frame dynamic mode decomposition includes:

[0008] Step S1: Obtain three phase-shifted interferograms of the object under test, and perform differential and normalization processing on the three phase-shifted interferograms to obtain the first interferogram sequence after removing the background;

[0009] Step S2: Expand the first interferogram sequence into column vector form and construct data matrix X and data matrix Y in chronological order;

[0010] Step S3: Perform singular value decomposition on the data matrix X to calculate the approximate matrix. ;

[0011] Step S4, for the approximate matrix Feature analysis is performed to extract eigenvalues ​​and eigenvectors, and the eigenvectors are mapped back to the space of the first interferogram sequence to reconstruct the dynamic pattern of the first interferogram sequence. ;

[0012] Step S5, according to the dynamic mode The phase angle information of the dominant mode is used to recover the phase distribution of the object under test.

[0013] Furthermore, step S1 includes:

[0014] The expression for the light intensity of a three-frame phase-shifted interferogram is:

[0015] ;

[0016] , , ;

[0017] in, Let K be the light intensity at index k in the nth frame of the interferogram; n is the frame number of the interferogram; k is the position index of the pixel in the interferogram; and K is the total number of pixels in the interferogram. Background light intensity; For modulation amplitude; The phase term to be measured; Let n be the phase shift step size of the nth frame of the interferogram;

[0018] The differential operation is performed on three phase-shifted interferograms, and the expression is:

[0019] ;

[0020] ;

[0021] ;

[0022] In the formula, This is a difference graph; ;

[0023] The L2 norm of a three-frame phase-shifted interferogram is calculated using the following expression:

[0024] ; ; ;

[0025] In the formula, , , It is an L2 norm;

[0026] Based on the L2 norm, the three-frame phase-shifted interferogram is normalized as follows:

[0027] ;

[0028] ;

[0029] ;

[0030] In the formula, ;

[0031] Introducing global constants And redefine the new modulation amplitude term ;

[0032] The first interferogram sequence is obtained, and its expression is:

[0033] ;

[0034] ;

[0035] ;

[0036] In the formula, This represents the value of the k-th pixel in the n-th frame of the interferogram after background removal. .

[0037] Furthermore, step S2 includes:

[0038] The first interferogram sequence is represented as follows:

[0039] ;

[0040] In the formula, is the intensity of the interferogram in the nth frame; n is the frame number of the interferogram; b is the modulation amplitude of the interferogram; denoted as the object phase of the object under test; 'a' is the phase shift step size.

[0041] The Euler transform of the first interferogram sequence is expressed as follows:

[0042] ;

[0043] In the formula, The phase factor for the positive frequency component of the interferogram; The phase factor for the negative frequency component of the interferogram; ω is the angular frequency, representing the phase shift step introduced in each frame; A positive frequency carrier indicates that the frequency changes with the number of frames n at an angular frequency. A complex signal rotating counterclockwise; A negative frequency carrier indicates that the frequency changes with the number of frames n at an angular frequency. A complex signal rotating clockwise;

[0044] The frequency domain expression obtained by performing a Fourier transform on the first interferogram sequence is as follows:

[0045] ;

[0046] In the formula, This is the frequency domain signal after Fourier transform; For a unit pulse; and For spatial modes with different oscillation frequencies, the oscillation frequencies are respectively and ;

[0047] The size of each interferogram in the first interferogram sequence is set to be... The interferograms in matrix form of each frame are rearranged into column vectors in column order. ;

[0048] The first interferogram sequence is divided into a data matrix. and data matrix The expression is:

[0049] ;

[0050] In the formula, The column vector of each frame of the interferogram N is the number of frames in the first interferogram sequence; Y is the interferogram data matrix for the first N-1 frames; Y is the interferogram data matrix for the last N-1 frames.

[0051] Furthermore, step S3 includes:

[0052] Selecting a data matrix X, we perform singular value decomposition, expressed as:

[0053] ;

[0054] In the formula, U represents a left singular vector matrix of size M×(N-1); Σ represents a right singular vector matrix of size (N-1)×M; Σ represents a singular value matrix of size (N-1)×(N-1); A right singular vector matrix The conjugate transpose of ; M is the total number of pixels in each interferogram in the data matrix X; N is the total number of frames in the interferograms of the data matrix X;

[0055] Constructing a linear dynamic model An approximate matrix is ​​constructed using the singular value decomposition results of the data matrix X. :

[0056] ;

[0057] In the formula, It is the conjugate transpose of the left singular vector matrix U; It is a right singular vector matrix; Let Σ be the transpose of the singular value matrix Σ; A is an M×M linear dynamic operator.

[0058] Further, step S4 includes:

[0059] For approximate matrices Eigenvalue decomposition is performed to obtain eigenvalues ​​and eigenvectors, expressed as:

[0060] ;

[0061] In the formula, For eigenvalues; For feature vectors;

[0062] eigenvectors Projecting back into the space of the first interferogram sequence yields the dynamic mode of the first interferogram sequence. The expression is:

[0063] ;

[0064] In the formula, Let be the left singular vector matrix of the first interferogram sequence.

[0065] Furthermore, step S5 includes:

[0066] choose The mode corresponding to the positive frequency component is denoted as . ;in For dynamic mode The first two columns are used as the dominant mode;

[0067] calculate The phase distribution of the measured object is recovered from the argument information, expressed as:

[0068] ;

[0069] In the formula, To recover the package phase map; It is the arctangent function in the four quadrants; For dynamic mode Positive frequency dynamic mode; for The real part; for The imaginary part.

[0070] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0071] This invention discloses a three-frame dynamic mode decomposition method for in-situ phase-shift interferometry measurement. The method acquires three frames of interferograms with equal-step phase shifts, constructs a new phase-shift interferogram sequence through differential and normalization operations to remove background components and unify the modulation. A dynamic mode decomposition strategy is employed to decompose the newly constructed interferogram sequence, extracting its dominant dynamic mode to achieve high-precision phase reconstruction. This method does not rely on any iterative process, ensuring phase recovery accuracy while improving computational efficiency. It also exhibits good robustness to background intensity and modulation fluctuations and noise interference, making it suitable for the high-speed, real-time measurement requirements of integrated circuit production lines.

[0072] The method of this invention achieves a breakthrough in frame count compared to previous dynamic pattern decomposition algorithms, combining high precision and high efficiency. Attached Figure Description

[0073] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0074] Figure 1 This is a schematic flowchart of the method of the present invention;

[0075] Figure 2 The figures show the interferogram and corresponding reference phase distribution obtained through simulation in this invention. In the figures, image a is the reference phase distribution, and the color bars represent the phase values ​​at the corresponding pixels, in rad; image b is the interferogram, and the color bars represent the light intensity at the corresponding pixels, in W / m².

[0076] Figure 3The phase distribution results extracted under the simulation conditions of the method of the present invention are shown in the figure. In the figure, image a is the phase distribution, and the color bar is the phase value at the corresponding pixel point, in rad; image b is the error map corresponding to the phase, where the x-axis and y-axis are the pixel coordinates of the image, in pixels, and the z-axis and color bar are the error values ​​at the corresponding pixel points, in rad.

[0077] Figure 4 The error map corresponding to the phase extracted by the method of the present invention under different background intensity and modulation conditions is shown in the figure: In the figure, the x-axis and y-axis are the pixel coordinates of the image, in pixels, and the z-axis and color bar are the error values ​​at the corresponding pixels, in rad.

[0078] Figure 5 This is a schematic diagram of the optical path system used in this invention; in the figure, 1 is a laser; 2 is a first lens; 3 is a pinhole; 4 is a second lens; 5 is a polarizer; 6 is a beam splitter; 7 is a sample; 8 is a liquid crystal spatial light modulator; and 9 is a CCD camera.

[0079] Figure 6 The phase distribution and phase error maps extracted under the experimental conditions of the method of the present invention are shown in the figures: In the figures, images a1 and b1 are the phase distribution results, where the color bars are the phase values ​​at the corresponding pixels, in rad; images a2 and b2 are the phase error maps, where the x-axis and y-axis are the pixel coordinates of the images, in pixels, and the z-axis and color bars are the error values ​​at the corresponding pixels, in rad. Detailed Implementation

[0080] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0081] The purpose of this invention is to provide a phase-shift interferometric in-situ measurement method for three-frame dynamic mode decomposition, which aims to solve or improve at least one of the above-mentioned technical problems.

[0082] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0083] Definitions:

[0084] MATLAB: Signal Processing Toolbox is a powerful professional toolset that provides comprehensive support for signal generation, analysis, filtering, transformation, and visualization.

[0085] The awgn function is a core function in the MATLAB Signal Processing Toolbox. Its main function is to add Gaussian white noise to the input signal to simulate the unavoidable random noise interference in real communication or measurement systems.

[0086] Peaks: is a built-in example function in the MATLAB Signal Processing Toolbox. It generates a mathematical function with multiple local maxima and minima. It is often used to demonstrate graphical and numerical functions such as 3D plotting, surface fitting, interpolation, and optimization.

[0087] like Figure 1 As shown, this invention provides a phase-shift interferometric in-situ measurement method for three-frame dynamic mode decomposition, comprising:

[0088] Step S1: Obtain three phase-shifted interferograms of the object under test, and perform differential and normalization processing on the three phase-shifted interferograms to obtain the first interferogram sequence after removing the background. This includes the following steps:

[0089] Step S1-1, the expression for the light intensity of the three-frame phase-shifted interferogram is:

[0090] ;

[0091] , , ;

[0092] in, Let K be the light intensity at index k in the nth frame of the interferogram; n is the frame number of the interferogram; k is the position index of the pixel in the interferogram; and K is the total number of pixels in the interferogram. Background light intensity; For modulation amplitude; The phase term to be measured; Let be the phase shift step size of the nth frame of the interferogram.

[0093] Step S1-2: Perform a differential operation on the three-frame phase-shifted interferogram to remove the background component. The expression is:

[0094] ;

[0095] ;

[0096] ;

[0097] In the formula, This is a difference graph; .

[0098] Steps S1-3 involve normalizing the three phase-shifted interferograms, including the following steps:

[0099] The L2 norm of a three-frame phase-shifted interferogram is calculated using the following expression:

[0100] ; ; ;

[0101] In the formula, , , It is an L2 norm;

[0102] Based on the L2 norm, a normalization operation is performed on the three-frame phase-shifted interferogram, and the expression is:

[0103] ;

[0104] ;

[0105] ;

[0106] In the formula, ;

[0107] When the interference fringes in the three-frame phase-shifted interferogram are greater than 1, the following approximate values ​​exist:

[0108]

[0109] Introducing global constants And redefine the new modulation amplitude term ;

[0110] Constructing the first interferogram sequence with equal step size The expression is:

[0111] ;

[0112] ;

[0113] ;

[0114] In the formula, This represents the value of the k-th pixel in the n-th frame of the interferogram after background removal. .

[0115] Step S2 involves expanding the first interferogram sequence into a column vector form and constructing data matrices X and Y in chronological order, including the following steps:

[0116] The first interferogram sequence is represented as follows:

[0117] ;

[0118] In the formula, is the intensity of the interferogram in the nth frame; n is the frame number of the interferogram; b is the modulation amplitude of the interferogram; denoted as the object-light phase of the object under test; 'a' is the phase shift step size.

[0119] The Euler transform of the first interferogram sequence is expressed as follows:

[0120] ;

[0121] In the formula, The phase factor for the positive frequency component of the interferogram; The phase factor for the negative frequency component of the interferogram; ω is the angular frequency, representing the phase shift step introduced in each frame; A positive frequency carrier indicates that the frequency changes with the number of frames n at an angular frequency. A complex signal rotating counterclockwise; A negative frequency carrier indicates that the frequency changes with the number of frames n at an angular frequency. A complex signal rotating clockwise.

[0122] The frequency domain expression obtained by performing a Fourier transform on the first interferogram sequence is as follows:

[0123] ;

[0124] In the formula, This is the frequency domain signal after Fourier transform; For a unit pulse; and For spatial modes with different oscillation frequencies, the oscillation frequencies are respectively and .

[0125] Dynamic mode decomposition can be used to extract spatial modes with different oscillation frequencies from interferograms. Phase can be derived from the spatial mode. The corresponding oscillation frequency is obtained from this. This provides information on the phase shift step size.

[0126] The size of each interferogram in the first interferogram sequence is set to be... The interferograms in matrix form of each frame are rearranged into column vectors in column order. .

[0127] The first interferogram sequence is divided into a data matrix. and data matrix The expression is:

[0128] ; ;

[0129] In the formula, The column vector of each frame of the interferogram N is the number of frames in the first interferogram sequence; Y is the interferogram data matrix for the first N-1 frames; Y is the interferogram data matrix for the last N-1 frames.

[0130] Step S3: Perform singular value decomposition on the data matrix X and calculate the approximate matrix. ;

[0131] Selecting a data matrix X, we perform singular value decomposition, expressed as:

[0132] ;

[0133] In the formula, U represents a left singular vector matrix of size M×(N-1); Σ represents a right singular vector matrix of size (N-1)×M; Σ represents a singular value matrix of size (N-1)×(N-1); A right singular vector matrix The conjugate transpose of ; M is the total number of pixels in each interferogram in the data matrix X; N is the total number of frames in the interferograms in the data matrix X.

[0134] Constructing a linear dynamic model An approximate matrix is ​​constructed using the singular value decomposition results of the data matrix X. As a low-dimensional operator The estimate is expressed as:

[0135] ;

[0136] In the formula, It is the conjugate transpose of the left singular vector matrix U; It is a right singular vector matrix; Let Σ be the transpose of the singular value matrix Σ; A is an M×M linear dynamic operator.

[0137] Step S4, for the approximate matrix Feature analysis is performed to extract eigenvalues ​​and eigenvectors, and the eigenvectors are mapped back to the space of the first interferogram sequence to reconstruct the dynamic pattern of the first interferogram sequence. ;

[0138] For approximate matrices Eigenvalue decomposition is performed to obtain eigenvalues ​​and eigenvectors, expressed as:

[0139] ;

[0140] In the formula, For eigenvalues; These are the eigenvectors.

[0141] eigenvectors Projecting back into the space of the first interferogram sequence yields the dynamic mode of the first interferogram sequence. The expression is:

[0142] ;

[0143] Step S5, according to the dynamic mode The phase angle information of the dominant mode is used to recover the phase distribution of the object under test. The object under test refers to the target or sample undergoing phase distribution measurement.

[0144] The background component of the first interferogram sequence has been removed through a differential operation, dynamic mode. The first two columns Corresponding to its primary mode. Selection. The mode corresponding to the positive frequency component is denoted as . Then by calculation The phase distribution of the measured object is recovered from the argument information, expressed as:

[0145] ;

[0146] In the formula, To recover the package phase map; It is the arctangent function in the four quadrants; For dynamic mode Positive frequency dynamic mode; for The real part; for The imaginary part.

[0147] Example 1

[0148] The simulation generates an interference fringe pattern. The expressions for the fringe pattern background and modulation degree are as follows:

[0149] ;

[0150] ;

[0151] ;

[0152] In the formula, -1≤x≤1, -1≤y≤1, the phase shifts of the three interferograms are set to 0 rad, 1 rad, and 2 rad, respectively, and white Gaussian noise with a signal-to-noise ratio of 45 dB, generated by the awgn function in MATLAB, is added to the interferograms. Figure 2 Figure a shows the phase distribution of the theoretical reference, as shown in Figure 2. Figure 2 Figure b shows one frame of the interferogram.

[0153] To verify the accuracy and efficiency of the method of the present invention, the root mean square error (RMS) of the difference between the estimated phase distribution and the reference phase distribution is selected as a quantitative evaluation index.

[0154] The demodulated phase distribution is as follows Figure 3 As shown in Figure a, the difference between the estimated phase distribution and the reference phase distribution is as follows: Figure 3 As shown in b. Compare. Figure 3 a and Figure 2 In step a, the obtained phase distribution result is consistent with the reference phase. Figure 3 The value of b indicates that the error parameter remains within a very small range.

[0155] Table 1 shows the calculation results, including the root mean square error (RMS) of the phase distribution and the processing time. The RMS error of the phase distribution is 0.0195 rad, and the calculation time is 0.0191 s. This verifies the accuracy of the method of the present invention, and the processing time of the method is in the tens of milliseconds range, verifying the high efficiency of the method of the present invention.

[0156] Table 1. Demodulation phase root mean square error and computation time under simulation conditions

[0157]

[0158] Example 2

[0159] To verify the robustness of the method of this invention to background fluctuations, numerical simulation tests were conducted on fluctuations with different background intensities and modulation levels. The background and modulation level parameters were set as follows:

[0160] ;

[0161] ;

[0162] In the formula, -1≤x≤1, -1≤y≤1, and These represent background intensity and modulation amplitude, respectively. and It is a spatial attenuation parameter.

[0163] The phase distribution and phase shift of the interference fringes are consistent with those of Example 1, and 45 dB of white Gaussian noise is also added.

[0164] The simulation conditions are set as shown in Table 2, and the corresponding root mean square error (RMS) of the difference between the estimated phase distribution and the reference phase distribution is given. The difference between the demodulated estimated phase distribution and the reference phase distribution is as follows: Figure 4 As shown. In Figure 4In the table, a corresponds to case 1 in Table 2, b corresponds to case 2 in Table 2, c corresponds to case 3 in Table 2, and d corresponds to case 4 in Table 2.

[0165] Depend on Figure 4 It was found that fluctuations in background intensity and modulation had little impact on the root mean square error, verifying that the method of the present invention is robust to fluctuations in background intensity and modulation.

[0166] Table 2. Root mean square error of demodulation phase corresponding to different background and modulation fluctuations under simulation conditions.

[0167]

[0168] Example 3

[0169] To verify the feasibility and accuracy of the method of the present invention at the experimental level, the following methods were employed: Figure 5 The Michelson interferometer system is shown. The reference arm uses a liquid crystal spatial light modulator to generate the desired phase shift. The test arm uses a convex lens and a wafer surface as the test objects to obtain two different phase distributions with spherical and planar characteristics, corresponding to cases 1 and 2 in Table 3, respectively.

[0170] The working process of the Michelson structure interferometer system is as follows: Laser 1 emits a laser beam, which is modulated into a uniform Gaussian beam by the first lens 2, pinhole 3, and second lens 4. This beam is then modulated into linearly polarized light by the polarizer 5 and enters the beam splitter 6, resulting in two beams. One beam, a reference beam, illuminates the liquid crystal spatial light modulator 8 for wavefront modulation, while the other beam, a test beam, illuminates the surface to be tested on the sample 7. Finally, the two beams are reflected back to the beam splitter 6, combined to form interference fringes. These interference fringes are captured by the CCD camera 9 and transmitted to the data processing terminal.

[0171] During measurement, different grayscale phase masks are loaded onto the SLM synchronously to achieve equal-step phase shift control of the interferogram. Three phase-shifted interferograms are acquired by the camera and transmitted to the computer terminal to verify the three-frame phase demodulation effect of the method of the present invention at the experimental level.

[0172] In the two sets of experiments, the reconstructed phase distributions were as follows: Figure 6 a1 and Figure 6 As shown in b1, the error plots are as follows: Figure 6 a2 and Figure 6 As shown in b2.

[0173] Table 3 shows the corresponding root mean square error of the phase and the corresponding processing time. Figure 6As shown in Table 3, the root mean square error (RMS) of the phase distribution in the two sets of experimental data are 0.1295 rad and 0.0589 rad, respectively, and the processing times are 0.0073 s and 0.0065 s, respectively, which are similar to the simulation results.

[0174] Therefore, the method of the present invention has high measurement accuracy. While maintaining high accuracy, the calculation time for a single phase demodulation is only a few milliseconds, less than ten milliseconds, verifying the high accuracy and high efficiency of the method of the present invention.

[0175] Table 3. Root mean square error of demodulation phase and computation time of the invented method under experimental conditions.

[0176]

[0177] The above provides a specific example illustrating the in-situ phase-shift interferometric measurement method of the present invention based on a three-frame dynamic mode decomposition. In the technical implementation of the present invention, a liquid crystal spatial light modulator is used to adjust the optical field, thereby generating a phase shift, which in turn obtains a sequence of equal-step phase-shift interferograms and completes the phase recovery of the test target. This specific experimental platform and implementation method are intended to help those skilled in the art to more clearly understand the core ideas and operational procedures of the present invention.

[0178] It should be noted that the grouping strategy and differential normalization form of the three-frame interferograms can be flexibly adjusted according to different application requirements and interferogram acquisition methods. Without changing the basic principles of this invention, any equivalent improvements, functional enhancements, or specific structural adaptations made to this method should be considered as part of the technical solution of this invention and fall within the protection scope of the appended claims.

[0179] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0180] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A phase-shift interferometric in-situ measurement method for three-frame dynamic mode decomposition, characterized in that, include: Step S1: Obtain three phase-shifted interferograms of the object under test, and perform differential and normalization processing on the three phase-shifted interferograms to obtain the first interferogram sequence after removing the background; Step S2: Expand the first interferogram sequence into column vector form, and construct data matrix X and data matrix Y in chronological order; Step S3: Perform singular value decomposition on the data matrix X to calculate an approximate matrix. ; Step S4, for the approximate matrix Feature analysis is performed to extract eigenvalues ​​and eigenvectors, and the eigenvectors are mapped back to the space of the first interferogram sequence to reconstruct the dynamic pattern of the first interferogram sequence. ; Step S5, according to the dynamic mode The phase angle information of the dominant mode is used to recover the phase distribution of the object under test; Step S4 includes: For the approximate matrix Eigenvalue decomposition is performed to obtain eigenvalues ​​and eigenvectors, expressed as: ； In the formula, For eigenvalues; For feature vectors; The feature vector Projecting back into the space of the first interferogram sequence, the dynamic mode of the first interferogram sequence is obtained. The expression is: ； In the formula, Let be the left singular vector matrix of the first interferogram sequence; Step S5 includes: choose The mode corresponding to the positive frequency component is denoted as . ;in For the dynamic mode The first two columns are used as the dominant mode; Calculate the The phase distribution of the object under test is recovered from the argument information, expressed as: ； In the formula, To recover the package phase map; It is the arctangent function in the four quadrants; For dynamic mode Positive frequency dynamic mode; for The real part; for The imaginary part.

2. The in-situ phase-shift interferometric measurement method for three-frame dynamic mode decomposition according to claim 1, characterized in that, Step S1 includes: The light intensity expression for the three-frame phase-shifted interferogram is: ； ， ， ； in, Let K be the light intensity at index k in the nth frame of the interferogram; n is the frame number of the interferogram; k is the position index of the pixel in the interferogram; and K is the total number of pixels in the interferogram. Background light intensity; For modulation amplitude; The phase term to be measured; Let n be the phase shift step size of the nth frame of the interferogram; The differential operation is performed on the three phase-shifted interferograms, and the expression is: ； ； ； In the formula, This is a difference graph; ; The L2 norm of the three-frame phase-shifted interferogram is calculated using the following expression: ； ； ； In the formula, , , It is an L2 norm; Based on the L2 norm, the three-frame phase-shift interferogram is normalized as follows: ； ； ； In the formula, ; Introducing global constants And redefine the new modulation amplitude term ; The first interferogram sequence is obtained, and its expression is: ； ； ； In the formula, This represents the value of the k-th pixel in the n-th frame of the interferogram after background removal. .

3. The in-situ phase-shift interferometric measurement method for three-frame dynamic mode decomposition according to claim 1, characterized in that, Step S2 includes: The first interferogram sequence is represented as follows: ； In the formula, is the intensity of the interferogram in the nth frame; n is the frame number of the interferogram; b is the modulation amplitude of the interferogram; denoted as the object phase of the object under test; 'a' is the phase shift step size. The Euler transform of the first interferogram sequence is expressed as follows: ； In the formula, The phase factor for the positive frequency component of the interferogram; The phase factor for the negative frequency component of the interferogram; ω is the angular frequency, representing the phase shift step introduced in each frame; A positive frequency carrier indicates that the frequency changes with the number of frames n at an angular frequency. A complex signal rotating counterclockwise; A negative frequency carrier indicates that the frequency changes with the number of frames n at an angular frequency. A complex signal rotating clockwise; Performing a Fourier transform on the first interferogram sequence yields the following frequency domain expression: ； In the formula, The frequency domain signal is the result of the Fourier transform; For a unit pulse; and For spatial modes with different oscillation frequencies, the oscillation frequencies are respectively and ; The size of each interferogram in the first interferogram sequence is set to be... The interferograms in matrix form of each frame are rearranged into column vectors in column order. ; The first interferogram sequence is divided into a data matrix. and data matrix The expression is: ； In the formula, The column vector of each frame of the interferogram N is the number of frames in the first interferogram sequence; Y is the interferogram data matrix for the first N-1 frames; Y is the interferogram data matrix for the last N-1 frames.

4. The in-situ phase-shift interferometric measurement method for three-frame dynamic mode decomposition according to claim 1, characterized in that, Step S3 includes: Select the data matrix X and perform singular value decomposition, the expression of which is: ； In the formula, U represents a left singular vector matrix of size M×(N-1); Σ represents a right singular vector matrix of size (N-1)×M; Σ represents a singular value matrix of size (N-1)×(N-1); A right singular vector matrix The conjugate transpose of ; M is the total number of pixels in each interferogram in the data matrix X; N is the total number of frames in the interferograms of the data matrix X; Constructing a linear dynamic model An approximate matrix is ​​constructed using the singular value decomposition results of the data matrix X. : ； In the formula, It is the conjugate transpose of the left singular vector matrix U; It is a right singular vector matrix; Let Σ be the transpose of the singular value matrix Σ; A is an M×M linear dynamic operator.

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