Optical interferometric phase extraction method based on principal component analysis and vector inverse orthogonalization

Abstract

An optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization, which regards the light intensity values of the same pixel points in interference images as a multi-dimensional light intensity vector, according to the double-beam interference light intensity formula, regards the light intensity vectors of different pixel points as the result of linear representation by a group of unknown initial basis vectors, solves the real phase shift between the interference images by solving the unknown initial basis vectors, and finally solves the wrapped phase of the object surface height; when calculating, firstly, a group of standard orthogonal bases of the interference light intensity vector is solved by the principal component analysis method, and the first two principal components of the standard orthogonal bases are considered as the result of orthogonalization of the unknown initial basis vectors; then the constraint conditions are set to establish the equation group, the initial basis vectors are calculated by vector inverse orthogonalization, so as to calculate the real phase shift between the interference images, substitute into the linear over-determined equation group constructed according to the double-beam interference light intensity theory, and solve the least square solution of the wrapped phase of the object surface height.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of optical precision measurement, and relates to optical phase shifting interferometry, in particular to an optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization. BACKGROUND

[0002] Optical phase shifting interferometry is a precision measurement method based on the principle of light wave interference, which is widely used in industrial manufacturing and medical diagnosis. The basic principle is that when two coherent beams meet at a certain point in space, the phase relationship of coherent light waves will cause changes in light intensity, thereby forming an interference pattern. The core of optical phase shifting interferometry is to use optical phase extraction technology to calculate the height phase of the measured object surface.

[0003] Traditional phase shifting interferometry phase extraction technology requires a fixed step size for single phase shifting, while current interferometric microscopy systems generally use a miniature piezoelectric displacement device installed on an interference objective lens to perform interference phase shifting. Such miniature piezoelectric displacement devices generally have several to tens of nanometers of motion error during movement, resulting in phase extraction calculation error. In recent years, there have been two major trends in optical phase shifting interferometry phase extraction technology. One is to reduce the number of interference images involved in the calculation to improve system measurement efficiency. Such algorithms include the two-step phase shifting algorithm of Gram-Schmidt orthogonalization. This type of method is fast, but sensitive to noise and has lower calculation accuracy than the multi-step phase shifting method, so it is only suitable for rapid online measurement of samples. The second is the emergence of multi-step blind phase shifting technology that does not require phase shifting. The most representative of this type of algorithm is the principal component analysis algorithm and the advanced iterative algorithm, which do not require iteration. Although the non-iterative principal component analysis phase extraction algorithm is fast, its calculation accuracy is often lower than that of the iterative method and it is difficult to determine the global phase sign of the measured object. Although the iterative algorithm can often obtain very accurate solutions, it takes a long time to solve multiple iterations, and when there are large background fluctuations and nonlinear fluctuations in the modulation degree in the interference image, the calculation error cannot be ignored. In addition, some algorithms also require a certain number of stripes in the interference image, and when the height phase distribution of the object surface is less than one period, there will be a larger calculation error, which is not conducive to the measurement of flat surfaces of the measured sample. Therefore, there is an urgent need for a multi-step blind phase shifting interferometry phase extraction technology that has high robustness, does not require a certain number of stripes in the interference image, and has both calculation speed and accuracy.

[0004] The above technical problem is the main problem faced by current optical interference phase extraction technology. SUMMARY

[0005] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide an optical interferometry phase extraction method (PCAVR) based on principal component analysis and vector inverse orthogonalization to solve the problems of current optical interferometry phase extraction technology such as low accuracy, easy to produce errors, and long time consumption.

[0006] To achieve this objective, the present invention treats the light intensity value at a single pixel in an interference pattern as a multidimensional light intensity vector linearly represented by a set of unknown initial basis vectors. The actual phase shift between the unknown initial basis vectors and the interference pattern is a trigonometric function. Principal component analysis and vector inverse orthogonalization are used to determine the unknown initial basis vectors, which are then used to calculate the phase shift between the interference patterns and the wrapped phase at the object's surface height.

[0007] The specific technical solutions of the present invention can be described as follows:

[0008] An optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization includes the following steps:

[0009] Step 1: reconstruct the interference image data according to the requirements of the principal component analysis method to obtain the light intensity matrix I, and the number of interference images is at least 4;

[0010] Step 2: Calculate the covariance matrix C of the light intensity matrix I and perform principal component analysis to obtain the two principal element vectors p1 and p2 corresponding to the two largest eigenvalues ​​λ1 and λ2;

[0011] Step 3: According to the constraints of the basis vectors, the orthogonal principal element vectors p1 and p2 are deorthogonalized to solve the initial basis vectors V1 and V2.

[0012] Step 4: Perform arctangent operation on the initial basis vectors V1 and V2 to obtain the wrapped phase shift between the interference images; perform one-dimensional phase unwrapping and phase flip alignment on the wrapped phase shift between the interference images to obtain the true phase shift δ between the interference images. n ;

[0013] Step 5: construct an overdetermined linear equation system containing the height phase information of the object to be measured, and obtain the surface height phase information of the object to be measured by solving the least squares solution;

[0014] Step 6: Analyze the eigenvalues ​​obtained by principal component analysis to determine whether the phase calculation result is reliable. If the calculation result is reliable, output the surface height phase information of the measured object and end the calculation;

[0015] Step 7: When the calculation result is judged to be unreliable, the light intensity matrix I is corrected using the solution obtained in step 5, and steps 2 to 6 are repeated for iterative calculation until the algorithm converges or the set maximum number of iterations is reached.

[0016] Compared with the prior art, the present application has the advantages of:

[0017] The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization of the present application is not only completely insensitive to the modulation amplitude fluctuation in the interference image, but also can judge the quality of the interference image through various thresholds, that is, the interference image used for phase extraction exists background light intensity fluctuation, and the influence of the background light intensity fluctuation between pixels in the interference image on the phase extraction can be eliminated through iteration to continuously correct the original interference image, so that the height phase information of the measured object can be accurately calculated. When the quality of the interference image is high and there is no obvious background light intensity fluctuation, the method of the present application does not need iteration, and the calculation time is only 20% of the AIA iteration algorithm. When the quality of the interference image is poor and there is severe background light intensity fluctuation, the method of the present application has smaller calculation error than the AIA algorithm.

[0018] In addition, the calculation amount of the method of the present application is concentrated on solving the nonlinear equation set of each pixel point in the interference image, and the calculation between the pixel points is completely independent in this calculation process. This characteristic makes the method of the present application extremely suitable for multi-core parallel operation, so that the present calculation time can be shortened by more than 75%.

[0019] When there is inevitable phase shift error in the optical phase shift interference system, the method of the present application is particularly suitable for precise measurement of measured samples with small microstructures or extremely high surface processing precision, such as micro-nano sensors, precision optical elements, semiconductor wafers, etc. BRIEF DESCRIPTION OF DRAWINGS

[0020] Figure 1 Flowchart of the optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization of the present application

[0021] Figure 2 Phase calculation results of the PCAVR algorithm, the PCA+LSM algorithm and the AIA algorithm of the present application under different degrees of background noise. Among them, (a) is an ideal interference image under 0% noise, (b) is an interference image under 15% background noise, (c) is the theoretical phase of a simulated inclined surface, and (d) is the residual of the PCAVR algorithm under 15% background noise.

[0022] Figure 3 Calculation time of the PCAVR algorithm, the PCA+LSM algorithm and the AIA algorithm of the present application under different degrees of background noise.

[0023] Figure 4The phase calculation results of the PCAVR algorithm, the PCA+LSM algorithm and the AIA algorithm of the application under the condition of 3% background noise and modulation degree variation. Wherein, (a) is the interference image added with 3% background noise and modulation degree variation, (b) is the phase residual of the PCAVR algorithm, (c) is the phase residual of the PCA+LSM algorithm, and (d) is the phase residual of the AIA algorithm.

[0024] Figure 5 The phase calculation results of the PCAVR algorithm, the PCA+LSM algorithm and the AIA algorithm of the application under the condition of 3% background noise and modulation degree variation. Wherein, (a) is the interference image added with 3% background noise and modulation degree variation, (b) is the phase residual of the PCAVR algorithm, (c) is the phase residual of the PCA+LSM algorithm, and (d) is the phase residual of the AIA algorithm. DETAILED DESCRIPTION

[0025] The embodiments of the application will be described in detail below with reference to the accompanying drawings and examples.

[0026] The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization of the application regards the light intensity values of the same pixel point in the interference images as a multi-dimensional light intensity vector, and regards the light intensity vectors of different pixel points as the results linearly represented by a group of unknown initial basis vectors according to the double-beam interference light intensity formula. The phase shift amount between the unknown initial basis vectors and the interference images is a strict trigonometric function relationship. The real phase shift amount between the interference images can be accurately calculated by solving the unknown initial basis vectors, and finally the wrapped phase of the object surface height is solved according to the obtained phase shift amount of the interference images. When calculating, firstly, a group of standard orthogonal bases of the interference light intensity vector is solved by the principal component analysis method, and the first two principal components of the standard orthogonal bases are regarded as the results after orthogonalization of the unknown initial basis vectors. Then, the constraint conditions are set according to the trigonometric function relationship of the initial basis vectors to establish an equation group, and the initial basis vectors are calculated by vector inverse orthogonalization, so as to calculate the real phase shift amount between the interference images. Finally, the real phase shift amount between the interference images is substituted into the linear over-determined equation group constructed according to the double-beam interference light intensity theory, and the least square solution of the wrapped phase of the object surface height is solved.

[0027] As shown in Figure 1 The method of the application mainly includes the following steps:

[0028] Step one, the interference images are reconstructed according to the requirements of the principal component analysis method to obtain a light intensity matrix I, wherein the number of the interference images is at least 4.

[0029] The phase-shift interference image set of the measured object of the present invention is collected by an optical phase-shift interferometer. The collected interference image should have good fringe contrast. On this basis, the interference pattern is reconstructed according to the requirements of the principal component analysis method to obtain the light intensity matrix I.

[0030] According to the principle of optical interference, the data reconstruction principle and specific method of this step are as follows: the grayscale value of each pixel in the phase-shift interference image is equal to the interference light intensity value. The interference light intensity values ​​of the same pixel in different interference images constitute an n-dimensional interference light intensity vector I(x, y), which is expressed as follows:

[0031] I(x,y)=[I1(x,y),I2(x,y),…,I n (x,y)]

[0032] Among them I n (x, y) represents the interference light intensity value of the pixel (x, y) in the nth interference image. The intensity matrix I is composed of the interference light intensity vectors of all pixels and is expressed as follows:

[0033] I=[I(1,1),I(1,2),I(1,3),…,I(N x ,N y )]

[0034] The matrix has a total of n rows N x ×N y Column, n is the number of interference patterns, N x ×N y is the resolution of the interference image. Furthermore, the interference light intensity value I of the pixel (x, y) in the nth interference image is n (x,y) is represented as:

[0035] I n (x,y)=A(x,y)+B(x,y)cos[Φ(x,y)+δ n ]

[0036] A(x,y) is the background light intensity at a pixel, B(x,y) is the modulation index of the interference signal at that pixel, and Φ(x,y) is the height phase of the object surface. Assuming there are n interference images in the interference image set, consider the light intensity values ​​of the same pixel in different interference images as a set of data. Performing a trigonometric expansion of the light intensity expression yields the following system of equations.

[0037]

[0038] The equations can be rewritten into vector form I(x,y)=α(x,y)V0+β(x,y)V1+γ(x,y)V2

[0039] where I(x, y) = [I1(x, y), I2(x, y),..., I n (x, y)] T , V0= [1, 1,..., 1] T , V1= [cosδ1, cosδ2,..., cosδ n ] T , V2= [sinδ1, sinδ2,..., sinδ n ] T , [T] is a transpose operation, according to the basic theory of optical interference, α(x, y) = A(x, y), β(x, y) = B(x, y)cosΦ(x, y), γ(x, y) = -B(x, y)sinΦ(x, y). A(x, y) and B(x, y) represent the background light intensity and modulation of the interference signal at pixel coordinates (x, y) respectively, and Φ(x, y) is the object surface height phase to be solved at the pixel. From the above formula, the vector I(x, y) formed by different pixels can be linearly represented by a group of initial basis vectors V0, V1, V2 independent of the object surface topography, and the initial basis vectors V1, V2 contain the real phase shift information between different interference images. Removing the background light intensity in the above expression can obtain At this time is completely linearly represented by the initial basis vectors V1, V2. The present application uses the principal component analysis method and the principal component vector inverse orthogonalization to solve the initial basis vectors V1, V2, so as to obtain the accurate phase shift between the interference images. Finally, the accurate phase shift solved is substituted into the over-determined linear equation set to obtain the real phase Φ(x, y) of the object surface.

[0040] Step two, calculate the covariance matrix C of the matrix I, C = (I-m x )(I-m x ) T , the matrix m x is the same size as the light intensity matrix I, and all elements in the matrix m x are equal and are the average of all elements of the light intensity matrix I. Principal component analysis is performed to obtain the principal component vectors, and the two principal component vectors p1, p2 corresponding to the largest two eigenvalues λ1, λ2 are taken.

[0041] The specific implementation method of the principal component analysis of the covariance matrix C is as follows:

[0042] (I-m x ) can perform centering processing on the data, which has a removing and inhibiting effect on the background light intensity, and all elements in the matrix m x are equal and are the average of all elements of the interference light intensity matrix I, while in other algorithms based on principal component analysis, the matrix m xAll elements in each row are equal and are the mean of all elements in the corresponding row of the interference intensity matrix I. This change is due to the fact that when the fringe period in the interference image is less than 1 or the number of fringes is small, one interference image may be darker overall, while the next interference image may be brighter overall. The background intensity calculated from a single interference image may have a large error from the actual value. Therefore, let the matrix m x All elements in are equal and are the mean of all elements of the interference intensity matrix I to better suppress the background light intensity. Since C is a real symmetric matrix, according to linear algebra theory, it is possible to find a transformation matrix D to diagonalize C, that is, F = DCD T , where D consists of the eigenvectors of the covariance matrix C, which are orthogonal to each other. F is the result of diagonalizing the C matrix, and the elements on its diagonal are the eigenvalues ​​of the C matrix. The matrix diagonalization calculation is completed using the singular value decomposition method and the Hotelling transform algorithm, and the solved eigenvalues ​​are recorded from large to small as λ1, λ2, ..., λ n , the eigenvalues ​​corresponding to the eigenvectors, i.e., principal element vectors, are denoted as p1, p2, ..., p n .

[0043] Step 3: According to the constraints of the basis vectors, perform vector inverse orthogonalization on the orthogonal principal element vectors p1 and p2 to solve the initial basis vectors V1 and V2.

[0044] The initial basis vectors V1 and V2 that are not strictly orthogonal and unknown can be solved by inverse orthogonalizing the orthogonal principal element vectors p1 and p2 under the constraints of trigonometric functions. The specific solution method for the initial basis vectors V1 and V2 in this step: Principal component analysis is essentially to find a set of standard orthogonal bases to replace the initial basis vectors to re-represent and reduce the dimension of the vector data. The standard orthogonal bases are linearly related to the initial basis vectors. In the calculation process of the covariance matrix C, the operation (Im x ) actually suppresses and removes the influence of background light intensity, namely α(x,y)V0. The difference in interference light intensity between pixels is determined by the surface phase Φ(x,y) of the object and the modulation degree B(x,y) of the interference signal. As can be seen from the previous analysis, the pixel vector of the interference pattern after removing the background light intensity is It can be regarded as a linear representation of a set of fixed initial basis vectors V1, V2. And it is considered that p1 and p2 are the results of orthogonalization of the initial basis vectors V1 and V2, so the initial basis vectors V1, V2 can be linearly represented by the first two principal components p1 and p2 solved by the principal component analysis method in step 2. According to the relationship between V1, V2 and p1, p2, the initial basis vector is expressed as Where k1, k2, k3, k4 are unknown coefficients to be solved, and they are solved using the least squares method. The values ​​of each dimension of the initial basis vectors V1 and V2 are the sine or cosine of the phase shift between the interference images, so the initial basis vectors satisfy the constraint V1(n) 2 +V2(n) 2 = 1, where V1(n) and V2(n) represent the values ​​of the initial basis vectors V1 and V2 in the nth dimension, respectively. Therefore, there are n constraint equations in total. Based on the constraint conditions, the following quadratic equation system can be obtained.

[0045]

[0046] The system of equations contains four unknowns, so this method requires four or more interference images to solve the quadratic equation. The following is the specific process of using the Levenberg-Marquard algorithm to solve the unknowns k1, k2, k3, and k4:

[0047] 3.1、Create vector function f(k), f(k)=[f1,f2,..,f n ] T ,k=[k1,k2,k3,k4] T And the error evaluation function F(k),

[0048]

[0049] 3.2. Given the initial iteration vector k (0) And the initial trust region radius u0, and start iteration.

[0050] u0=maxElement[J T (k (0) )J(k (0) )]

[0051] Where J(k) is the Jacobian matrix of the vector function f(k), and maxElement(x) means taking the maximum element of the matrix.

[0052]

[0053] 3.3. When iterating to the mth time, calculate the corresponding Jacobian matrix J[k (m) ] and the approximate Hessian matrix H[k (m) ]=J T [k (m) ]J[k (m) ].

[0054] 3.4 Solve the linear equations (H+μ m E)Δk=-J Tf, μ > 0, to obtain a step increment Δk, where E is an identity matrix.

[0055] 3.5, Calculate the gain ratio p and correct the trust region radius u for the next iteration m+1 .

[0056]

[0057] If p < 0.25, it means that the Taylor approximation is poor, and the trust region radius should be reduced, so u m+1 = 0.5u m .

[0058] If 0.25 < p < 0.75, it means that the Taylor approximation is moderate, and the current trust region radius should be maintained, so u m+1 = u m .

[0059] If p > 0.75, it means that the Taylor approximation is very accurate, and the trust region radius should be expanded, so u m+1 = 2u m .

[0060] 3.6, If p > 0, let k (m+1) = k (m) + Δk, otherwise let k (m+1) = k (m) .

[0061] 3.7, Determine whether the algorithm converges, if it does not converge or the maximum number of iterations given by the user is not reached, continue iteration, otherwise stop iteration and output the solution of k1, k2, k3, k4.

[0062] Step four, perform arctangent operation on the initial basis vectors V1, V2 to obtain the wrapped phase shift amount between the interference images. Then perform one-dimensional phase unwrapping and alignment flip operation on the wrapped phase shift amount between the interference images to obtain the real phase shift amount δ n .

[0063] This step first performs arctangent operation and one-dimensional phase unwrapping operation on the initial basis vectors V1, V2 to obtain the misaligned phase shift amount The formula is as follows:

[0064]

[0065] Where unwrap(x) represents one-dimensional phase unwrapping.

[0066] Subsequently, due to the uncertainty of the correspondence between V1 and V2, it is possible that and the real phase shift amount δ n are opposite numbers, so it is necessary to The global sign of is corrected, that is, the alignment flip operation is introduced. In the process of phase shift acquisition of interference pattern, the phase shift direction is determined, and it is customary to set the phase shift amount δ1 corresponding to the acquisition position of the first interference image to 0rad, so it is also necessary to set Aligned to 0rad. When the phase shift direction is positive, the actual phase value δ n according to Calculation, when the phase shift direction is negative, follow Calculate, where sgn(x) is the sign function, which is defined as follows:

[0067]

[0068] in, is the misaligned phase shift corresponding to the acquisition position of the second interference image. So far, the correct phase shift between the interference images and the global sign of the phase shift have been determined.

[0069] Step 5: Construct an overdetermined linear equation system containing the height phase information of the object to be measured, and find its least square solution to obtain the surface height phase information of the object to be measured.

[0070] The specific method for solving the surface height phase information of the measured object in this step is as follows: the vector expression of the pixel intensity in the interference pattern can be rewritten as the following overdetermined linear equation system containing the height phase information of the measured object:

[0071]

[0072] Among them, α(x,y), β(x,y), and γ(x,y) are the three unknown intermediate process quantities corresponding to the pixel point (x,y). Since the equation system contains three intermediate process unknown quantities and more than four equations, its least squares solution can be found. The residual function of the interference light intensity at the pixel point (x,y) is expressed as:

[0073]

[0074] The intensity of the k-th interference pattern calculated by solving the equation system is obtained by taking the partial derivative of the residual function and setting it to 0 to obtain the following solution:

[0075]

[0076] [α(x,y),β(x,y),γ(x,y)] T =L -1 b

[0077] in

[0078]

[0079] The above formula omits the pixel coordinates. The height phase value of each pixel is calculated by the inverse tangent trigonometric function, which is expressed as follows:

[0080]

[0081] Step 6: Analyze the eigenvalues ​​obtained by principal component analysis to determine whether the phase calculation result is reliable. If the calculation result is reliable, output the surface height phase information of the measured object and end the calculation.

[0082] The principle and method of judging the reliability of the phase calculation results in this step: When the background light intensity can be effectively suppressed by the mean, the intensity vector of the background light intensity is removed. It can be completely represented by the initial basis vectors V1, V2. For principal component analysis, λ2>>λ3, or λ3≈λ4≈…≈λ n Therefore, the reliable judgment basis for the phase calculation result is:

[0083]

[0084] or,

[0085]

[0086] If the above conditions are met, the phase calculation result is considered reliable, and the phase value calculated in step 5 can be directly output as the final result. ε1 and ε2 are user-defined thresholds. A larger ε1 and a smaller ε2 indicate a stricter reliability judgment and a smaller final calculation error. Generally, ε1 and ε2 are given as 3000 and 1.1, respectively.

[0087] Step 7: When the calculation result is judged to be unreliable, the light intensity matrix I is corrected using the solution in step 5, and steps 2 to 6 are repeated for iterative calculation until the PCAVR algorithm converges or reaches the set maximum number of iterations.

[0088] The specific correction method of the intensity vector matrix in this step is as follows: x ) effectively eliminates the background light intensity, the interference light intensity matrix is ​​corrected by α(x,y) solved in step 5 to more accurately remove the background light intensity. When the light intensity matrix I is in the mth iteration, its correction method is:

[0089] I (m) =I (1) -G (m-1)

[0090]

[0091] Among them I (m)is the correction result at the mth iteration, that is, the light intensity vector matrix participating in the principal component analysis at the mth iteration. (1) is the light intensity matrix obtained directly from the original interference image, α (m-1) (x, y) is the solution result of the unknown intermediate process quantity α(x, y) at the m-1th iteration, which is calculated in step 5.

[0092] The convergence judgment basis of the PCAVR algorithm of the present invention is: ε3 is the convergence threshold set by the user, which can be given as 0.003. is the phase shift vector of the i-th interferogram calculated at the m-th iteration, and the symbol |·| represents the absolute value.

[0093] In a specific embodiment, a set of simulated interference images was generated using Matlab software, and the proposed phase extraction method based on principal component analysis and vector inverse orthogonalization (PCAVR) was compared with the principal component analysis and least squares fusion algorithm (PCA+LSM) and the advanced iterative algorithm (AIA) to illustrate the superiority of the proposed PCAVR algorithm in robustness and solution accuracy.

[0094] The simulated interference image resolution is 1000 pixels × 1000 pixels, and the actual spatial size of each pixel is 0.345 nm × 0.345 nm. The measured surface is an ideal inclined plane with a z-height of z(x,y) = 0.345 x tan0.1 (in nm). The wavelength of the interference light source is 550 nm. The interference image set contains five interferograms. To simulate phase shift error, a random phase shift error of ±20 nm is added to the single phase shift of 50 nm. The resulting theoretical random phase shifts between the simulated interference images are 0 rad, 1.5133 rad, 1.9439 rad, 3.8050 rad, and 4.6906 rad, with varying phase shifts. Furthermore, to facilitate the display of the measured results and RMS calculation, a least-squares unwrapping algorithm based on discrete cosine transform (DCT) is used to unwrap the wrapped phases obtained using the PCAVR, PCA+LSM, and AIA algorithms. In addition, the convergence threshold of the AIA algorithm is set to 0.0005, the thresholds ε1, ε2, and ε3 of the PCAVR algorithm are set to 3000, 1.10, and 0.003, respectively, and the maximum number of iterations is set to 12.

[0095] Firstly, the four algorithms are compared in phase calculation by adding different levels of background noise to the ideal interferogram. The added background noise is Gaussian noise, and the standard deviation of the noise is determined by multiplying the standard deviation of the intensity of the interferogram set by a percentage. The background intensity A(x, y) of the pixels in the ideal interferogram is 120, and the modulation B(x, y) is 85. Table 1 shows the calculation results of the three algorithms, Figure 2 Fig. 1(a) is an ideal interferogram with 0% noise, Figure 2 Fig. 1(b) is an interferogram with 15% background noise, Figure 2 Fig. 1(c) is the theoretical phase of the simulated slope surface, Figure 2 Fig. 1(d) is the residual error of the PCAVR algorithm under 15% background noise. Figure 3 The calculation time of the three algorithms is shown. The results show that the PCA+LSM algorithm has the largest calculation error under the condition of only adding noise, even under the condition of 0% noise, there is still a phase shift calculation error of 0.0945 rad. Under the ideal 0 noise condition, the PCAVR algorithm can almost zero-error restore the real phase shift between the interferograms and the measured slope surface. Under various noise levels, the calculation error of the phase shift between the interferograms by the PCAVR algorithm is better than 10 -4 rad, which is one order of magnitude smaller than the AIA algorithm. When there is no background intensity fluctuation in the interferogram, the PCAVR algorithm can obtain accurate phase calculation results without iteration, while the AIA algorithm needs an average of 6 iterations to obtain the phase calculation results. In this case, the time consumption of the PCAVR algorithm is only 25% of that of the AIA algorithm. The PCAVR algorithm and the AIA algorithm are both accurate in calculating the phase shift between the interferograms, and the Gaussian noise is the main cause of the surface calculation error, which makes the performances of the two algorithms in RMS the same and better than that of the PCA+LSM algorithm.

[0096] Table 1 Calculation results of each algorithm under different levels of noise

[0097]

[0098] To illustrate the excellent performance of the PCAVR algorithm in dealing with the modulation fluctuation of the pixels in the interferogram, 3% background noise (signal-to-noise ratio of 30 Db) is added, and nonlinear modulation fluctuation is also added. The modulation of the central pixel of the interferogram is 85, and the modulation at the edge is 31.3. This fluctuation is huge, and the modulation of the pixel is calculated according to the following formula.

[0099]

[0100] Table 2 shows the calculation results of the three algorithms under the condition of 3% background noise and modulation fluctuation. Figure 4(a) shows the interference image with 3% background noise and modulation changes. Figure 4 (b) is the phase residual of the PCAVR algorithm. Figure 4 (c) is the phase residual of the PCA+LSM algorithm. Figure 4 (d) is the phase residual of the AIA algorithm. The calculation error and RMS of the phase shift between the interferograms of the PCAVR algorithm are 1.28×10 -4 rad and 0.0148, which are both smaller than the 0.0225rad and 0.0156 calculated by the AIA algorithm. The results show that the PCAVR algorithm outperforms the AIA algorithm in terms of both calculation speed and calculation error when the background modulation changes. The PCAVR algorithm is robust to the changes in the modulation between pixels.

[0101] Table 2 Calculation results of various algorithms under the conditions of 3% noise and modulation index change

[0102] Algorithm name PCAVR PCAVR+LSM AIA <![CDATA[Errorδ n (rad)]]> <![CDATA[1.28×10 -4 ]]> 0.0790 0.0225 RMS(rad) 0.0148 0.0258 0.0156 Time(s) 5.49 4.98 23.71

[0103] Finally, nonlinear background intensity and modulation changes are added to the original 3% background noise (SNR is 30dB). The background intensity and modulation changes of the pixels in the interference pattern are changed according to the following formulas:

[0104]

[0105] The maximum and minimum background intensity in the interference pattern are 120 and 44.1 respectively, and the maximum and minimum modulation index are 85 and 31.3 respectively. Table 3 shows the phase calculation results of the three algorithms under the conditions of adding 5% background noise and changing background intensity and modulation index. Figure 5 (a) shows the simulated interference image. Figure 5 (b) is the phase residual using the PCAVR algorithm, Figure 5 (c) is the phase residual using the PCA+LSM algorithm. Figure 5 Figure (d) shows the phase residual using the AIA algorithm. The results show that when the background light intensity in the interferogram fluctuates, the interferogram phase shift errors of the AIA and PCA+LSM algorithms are 0.1296 rad and 0.1439 rad, respectively, and the RMS values ​​are 0.0212 and 0.0219, respectively. However, the PCAVR algorithm can still accurately calculate the interferogram phase shift through iteration, with the calculated phase shift error and RMS between interferograms being 0.0065 rad and 0.0161, respectively.

[0106] Table 3 Calculation results of each algorithm under the conditions of 5% noise, background light intensity and modulation depth

[0107]

[0108]

[0109] The simulation result shows that the method can more accurately calculate the real phase shift between the interference images, accurately extract the phase information of the interference images collected with any phase shift step, and is not sensitive to the pixel modulation degree fluctuation in the interference image. When the pixel in the interference image has background light intensity fluctuation and modulation degree fluctuation, the calculation accuracy of the PCAVR algorithm is significantly better than that of the AIA algorithm and the PCA+LSM algorithm, and the PCAVR algorithm makes up for the deficiency of the AIA algorithm. Even if the interference image has severe background light intensity fluctuation, the PCAVR algorithm can accurately calculate the phase shift of the interference image and the height phase of the measured object.

[0110] When there is inevitable phase shift error in the optical phase shift interference device, the method is particularly suitable for precise measurement of the measured sample with small microstructure or extremely high processing precision, such as an optical element, a semiconductor wafer, and a micro-nano sensor. In the future, we plan to apply the algorithm to defect detection and surface profile measurement and roughness measurement of a wafer surface.

Claims

1. An optical interferometry phase extraction method based on principal component analysis and vector inverse orthogonalization, characterized in that: The steps include: Step 1: reconstruct the interference image data according to the requirements of the principal component analysis method to obtain the light intensity matrix I, and the number of interference images is at least 4; Step 2: Calculate the covariance matrix C of the light intensity matrix I and perform principal component analysis to obtain the two principal element vectors p1 and p2 corresponding to the two largest eigenvalues ​​λ1 and λ2; the covariance matrix C is expressed as follows: C=(I-m x )(I-m x ) T Among them, the matrix m x The same size as the intensity matrix I, the matrix m x All elements in are equal and are the mean of all elements of the intensity matrix I; Perform principal component analysis on the covariance matrix C to obtain the principal element vectors. The implementation method is as follows: The singular value decomposition method and Hotelling transform algorithm are used to complete the matrix diagonalization calculation, and the solved eigenvalues ​​are recorded as λ1, λ2, ..., λ from large to small. n , the eigenvalues ​​corresponding to the eigenvectors, i.e., principal element vectors, are denoted as p1, p2, ..., p n ; Step 3: According to the constraints of the basis vectors, perform vector inverse orthogonalization on the orthogonal principal element vectors p1 and p2 to solve the initial basis vectors V1 and V2; Step 4: Perform arctangent operation on the initial basis vectors V1 and V2 to obtain the wrapped phase shift between the interference images; perform one-dimensional phase unwrapping and flip alignment on the wrapped phase shift between the interference images to obtain the true phase shift δ between the interference images. n ; Step 5: construct an overdetermined linear equation system containing the height phase information of the object to be measured, and obtain the surface height phase information of the object to be measured by solving the least squares solution; Step 6: Analyze the eigenvalues ​​obtained by principal component analysis to determine whether the phase calculation result is reliable. If the calculation result is reliable, output the surface height phase information of the measured object and end the calculation; Step 7: If the calculation result is unreliable, the light intensity matrix I is corrected using the solution obtained in step 5, and steps 2 to 6 are repeated for iterative calculation until the algorithm converges or the set maximum number of iterations is reached; In step 1, a phase-shift interference image set of the object to be measured is collected by an optical phase-shift interferometer. The grayscale value of each pixel in the phase-shift interference image is equal to the interference light intensity value. The interference light intensity values ​​of the same pixel in different interference images constitute an n-dimensional interference light intensity vector I(x, y), which is expressed as follows: I(x,y)=[I1(x,y),I2(x,y),…,I n (x,y)] Among them I n (x, y) represents the interference light intensity value of the pixel (x, y) in the nth interference image. The intensity matrix I is composed of the interference light intensity vectors of all pixels and is expressed as follows: I=[I(1,1),I(1,2),I(1,3),…,I(N x ,N y )] The matrix has a total of n rows N x ×N y Column, n is the number of interference patterns, N x ×N y is the resolution of the interference image; The step 5 is to construct an overdetermined linear equation system containing the height phase information of the object being measured, which is expressed as follows: Among them, α(x,y), β(x,y), and γ(x,y) are three unknown intermediate process quantities corresponding to the pixel point (x,y). The residual function of the interference light intensity of the pixel point (x,y) is expressed as: Taking the partial derivative of the residual function and setting it to 0 gives the following solution: [α(x,y),β(x,y),γ(x,y)] T =L -1 b in The height phase value of each pixel is calculated by the inverse tangent trigonometric function, which is expressed as follows:

2. The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization according to claim 1 is characterized in that: In step 3, the constraints for solving the initial basis vectors V1 and V2 are: V1(n) 2 +V2(n) 2 =1 Among them, V1(n) and V2(n) represent the values ​​of the initial basis vectors V1 and V2 in the nth dimension respectively. The relationship between V1, V2 and p1, p2 is as follows: The unknown coefficients k1, k2, k3, and k4 are solved using the least squares method.

3. The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization according to claim 2 is characterized in that: In step 4, the initial basis vectors V1 and V2 are subjected to arc tangent operation and one-dimensional phase unwrapping operation to obtain the phase shift of the misaligned interference image. The formula is as follows: Here, unwrap(x) represents unwrapping the one-dimensional phase.

4. The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization according to claim 3 is characterized in that: The final phase shift is obtained by performing an alignment flip operation on the unaligned phase shift obtained by the one-dimensional phase unwrapping operation. The method is: Let the actual phase shift δ1 corresponding to the acquisition position of the first interference image be 0rad, then let Align to 0rad; When the phase shift direction is positive: When the phase shift direction is negative: Where sgn(x) is the sign function, which is defined as follows is the misaligned phase shift corresponding to the acquisition position of the second interference image.

5. The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization according to claim 1 is characterized in that: In step 6, the reliability of the phase calculation result is determined based on: or, Among them, ε1 and ε2 are the set thresholds.

6. The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization according to claim 5, characterized in that: In step seven, the light intensity matrix I is corrected in the mth iteration as follows: I (m) =I (1) -G (m-1) Among them I (m) is the correction result at the mth iteration, I (1) is the light intensity matrix obtained directly from the interference pattern, α (m-1) (x, y) is the solution result of the unknown intermediate process quantity α(x, y) at the m-1th iteration, which is calculated in step 5.

7. The optical interference phase extraction method based on principal component analysis and vector inverse orthogonalization according to claim 6, characterized in that: In step seven, the algorithm convergence judgment basis is Where ε3 is the set convergence threshold, is the phase shift vector of the i-th interferogram calculated at the m-th iteration, and the symbol |·| represents the absolute value.

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