A method for middle frequency surface shape white light interference splicing measurement based on polynomial filtering

Abstract

The application discloses a kind of based on polynomial filtering's intermediate frequency surface shape white light interference splicing measurement method, including to X ray plane mirror is carried out white light interference splicing measurement and obtains a group of adjacent sub-aperture spatial position overlapping interference measurement surface shape result;Using Legendre polynomial, the surface shape result of each sub-aperture is fitted to obtain polynomial coefficient under preset order;According to polynomial coefficient fitting, the fitting surface shape of corresponding order is generated, and all order fitting surface shapes are removed from measurement result together, obtain a group of after Legendre polynomial filtering surface shape result;After filtering surface shape result is spliced according to least square algorithm to obtain final measurement result.The present application aims to meet the sub-nanometer PV precision splicing measurement required in X ray plane mirror intermediate frequency surface shape error detection, using least square optimization splicing method and not being restricted by white light interferometer measurement error low frequency characteristics, improve the precision and flexibility of splicing measurement.

Description

Technical Field

[0001] This invention relates to the field of optical precision measurement technology, specifically to a mid-frequency surface shape white light interferometric splicing measurement method based on polynomial filtering. Background Technology

[0002] X-ray plane mirrors in fourth-generation synchrotron radiation sources and free-electron laser systems require a surface accuracy of better than 2 nm peak-to-valley (PV) height error and better than 50 nrad root mean square (RMS) slope error. Ion beam shaping (IBF) is a key process in the fabrication of X-ray plane mirrors, achieved through frequency-band combination processing. ~ The full-band error convergence, of which This is the length of the effective aperture of the X-ray plane mirror. (For...) ~ To improve the accuracy of slope error, IBF uses a small beam diameter processing method to modify the mid-frequency surface shape error, which relies on the high-precision measurement of mid-frequency surface shape error.

[0003] Currently, the main measurement technique used to guide the fabrication of X-ray mirrors is wavefront interferometry. The wavefront interferometry measurement results are used to guide the fabrication of X-ray plane mirrors. This is due to the wavefront interferometer's ability to detect the surface of X-ray mirrors. ~ High-precision measurement capability of low-frequency surface shape error across the frequency band. However, wavefront interferometers are limited by the instrument transfer function, and the repeatability and accuracy of mid-frequency surface shape error measurement cannot reach sub-nanometer PV accuracy. Therefore, it is difficult to achieve effective convergence of nanoradian accuracy slope error, i.e., mid-frequency error, using wavefront interferometry to guide IBF machining.

[0004] White light interferometers (WLIs) possess excellent mid-frequency response capabilities. Their high accuracy and repeatability in mid-frequency error measurement can effectively guide the mid-frequency correction stage in X-ray plane mirror frequency division processing. However, WLIs are limited by their measurement aperture, necessitating the use of sub-aperture stitching technology to expand the measurement range. Sub-aperture stitching-based metrology techniques have been widely adopted and extensively studied by numerous laboratories both domestically and internationally. Existing technologies have proposed and developed micro-stitching interferometry (MSI) using WLIs. While traditional stitching methods differ slightly in data processing, their core principles remain the same. The key lies in utilizing data from the overlapping regions of adjacent sub-apertures to optimize the tilt and piston errors between adjacent sub-apertures using the least squares method, achieving data alignment and fusion. This can be collectively referred to as the least squares optimization stitching (LSS) method. Since the algorithm optimization is based on data from the overlapping regions between adjacent sub-apertures, measurement errors present in each sub-aperture can affect the optimized tilt and piston errors, leading to inaccuracies in the stitching results. The main error component of white-light interferometry is low-frequency error, and the number of sub-apertures in MSI often reaches hundreds, leading to a severe error accumulation effect in the stitching results of the LSS method. Although existing methods have proposed angle monitoring and stitching methods that directly measure the tilt and piston error between adjacent sub-apertures using additional angle measuring devices, these require additional high-precision angle measuring instruments and complex supporting equipment. Furthermore, for ultra-high precision X-ray plane mirror inspection, the angle measurement accuracy generally needs to meet the stringent requirement of tens of nanoradians, which is difficult to achieve. Therefore, how to use a white-light interferometer to measure the mid-frequency surface shape error of an X-ray plane mirror with sub-nanometer PV accuracy is of great significance and has broad application prospects. Summary of the Invention

[0005] The technical problem to be solved by this invention is to provide a mid-frequency surface shape white light interferometric splicing measurement method based on polynomial filtering, which addresses the above-mentioned problems in the prior art. This invention aims to use a white light interferometer to achieve splicing measurement with sub-nanometer PV accuracy that meets the requirements for mid-frequency surface shape error detection of X-ray plane mirrors. It uses a least squares optimized splicing method and is not constrained by the low-frequency characteristics of white light interferometer measurement error, thereby improving the accuracy and flexibility of splicing measurement.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A mid-frequency surface shape white-light interferometric stitching measurement method based on polynomial filtering includes the following steps: performing white-light interferometric stitching measurement on an X-ray plane mirror to obtain a set of interferometric surface shape results where adjacent sub-apertures have overlapping spatial positions; fitting the surface shape result of each sub-aperture using Legendre polynomials to obtain a preset order of the surface shape result. The polynomial coefficients are as follows; based on the preset order. The polynomial coefficients are fitted to generate the corresponding order of fitted surface shapes, and all order fitted surface shapes are subtracted from the measurement results to obtain a set of surface shape results after Legendre polynomial filtering. The filtered surface shape results are then spliced ​​together using the least squares algorithm to obtain the final measurement results.

[0008] Optionally, the step of performing white-light interferometric stitching measurement on the X-ray plane mirror to obtain a set of interferometric surface shape results where the spatial positions of adjacent sub-apertures overlap includes:

[0009] S101: Based on the surface dimensions (Size) of the X-ray plane mirror being measured. x,y ) and the reference plane R of the white light interferometer microscope objective. x,y The entire surface T to be measured is divided into m sub-apertures, and the spatial positions of the m sub-apertures are arranged sequentially from left to right, with the sub-aperture spacing d satisfying the given overlap rate O.

[0010] S102: Position the X-ray plane mirror to the corresponding position on the stitching measurement system. Initially set the sub-aperture number i to 1. Drive the motion platform of the stitching system to move the mirror under test to the i-th sub-aperture T. i Location;

[0011] S103: The i-th sub-aperture T i Establish a local coordinate system L at the center i The origin is located at the i-th sub-aperture T. i The geometric center, Z i Along the i-th sub-aperture T i The normal arrangement drives the splicing system's electronically controlled two-dimensional adjustment table to adjust the number of interference fringes to zero;

[0012] S104: The piezoelectric ceramic of the white light interferometer drives the interference objective along the Z-axis. i The axial direction is scanned vertically based on a given scan step size s, and multiple interferograms are acquired in a single scan.

[0013] S105: The zero optical path difference point is extracted using a frequency domain algorithm to obtain the i-th sub-aperture T. i Interferometric measurement results M i ;

[0014] S106: Drive the motion platform of the stitching system to move the mirror under test to the (i+1)th sub-aperture T. i+1 The position is determined and the process jumps to step S103 until the surface shape results of all m sub-apertures are obtained.

[0015] Optionally, the surface shape result of each sub-aperture is fitted using Legendre polynomials to obtain a preset order of the surface shape result. When applying polynomial coefficients, the fitting of the surface shape results for each sub-aperture includes:

[0016] S201: Obtain the surface shape M to be analyzed i The discrete sampling matrix Z is used to record the effective pixel coordinates;

[0017] S202: Establish the surface result M i Using a two-dimensional Cartesian coordinate system with the image center at the origin and pixel center spacing in units, generate unnormalized coordinate vectors. and ;

[0018] S203: Construction and Analysis of Surface Shape M i Mask matrix Z of the same size mask Mark all elements without a value (NaN) as invalid points, and only retain the valid pixels corresponding to the valid pixel coordinates;

[0019] S204: Transform the mask matrix Z mask The unnormalized coordinate vector corresponding to the effective pixels and The surface values ​​in the discrete sampling matrix Z are recombined into one-dimensional vectors;

[0020] S205: Set the normalized width and height of the rectangular aperture, and map the surface matrix of the rectangular aperture to the normalized domain [-1,1]×[-1,1], thereby obtaining the normalized rectangular domain of the rectangular aperture;

[0021] S206: Construct a two-dimensional Legendre orthogonal polynomial basis matrix within a normalized rectangular domain. ;

[0022] S207: Based on the two-dimensional Legendre orthogonal polynomial basis matrix Constructing a least-squares linear system ,in Let the Legendre coefficient vector be... The effective surface shape magnitude vector;

[0023] S208: Solving least-squares linear systems using one-dimensional vectors Obtain the preset order The surface shape M to be analyzed below i The Legendre coefficient vector is used as the preset order for obtaining the surface shape result. The polynomial coefficients under the given conditions.

[0024] Optionally, the size of the discrete sampling matrix Z in step S201 is In step S202, an unnormalized coordinate vector is generated. and satisfy:

[0025] ;

[0026] ;

[0027] in, and Let Z be the number of rows and columns of the discrete sampling matrix.

[0028] Optionally, in step S206, a two-dimensional Legendre orthogonal polynomial basis matrix is ​​constructed within the normalized rectangular domain. When constructing the two-dimensional Legendre orthogonal polynomial basis matrix The dimension is Where N is the number of effective pixels, For the basis order, the basis order The value is set to the preset order. And the two-dimensional Legendre orthogonal polynomial basis matrix Any k-th column is given by the following formula:

[0029] ;

[0030] in, The basis matrix of the two-dimensional Legendre orthogonal polynomials For any k-th column, For about Two-dimensional substrate, They are respectively for Normalized coordinate vector, For about of A one-dimensional Legendre polynomial of order 1 For about of One-dimensional Legendre polynomial of order 1, order satisfy And under the same total order, according to Decreasing Arranged in ascending order, and having:

[0031] ;

[0032] in, and Let Z be the number of rows and columns of the discrete sampling matrix.

[0033] Optionally, the step of determining according to a preset order The polynomial coefficients are fitted to generate the corresponding order of fitted surface shapes, and all order fitted surface shapes are subtracted from the measurement results to obtain a set of surface shape results after Legendre polynomial filtering, including:

[0034] S301: Input indicates the preset order Legendre coefficient vector of polynomial coefficients ;

[0035] S302: Establish a two-dimensional grid coordinate matrix with the same size as the surface to be fitted and establish coordinate vectors with pixel centers;

[0036] S303: Set the normalization factor to map pixel coordinates to a normalized rectangular area;

[0037] S304: Construct a two-dimensional Legendre orthogonal polynomial basis matrix within a normalized rectangular domain. ;

[0038] S305: Calculate the discrete values ​​of the fitted surface shape. ,in For the Legendre coefficient vector; the discrete values ​​of the fitted surface shape Reshape into a fitted surface matrix of the same size as the original measurement mesh. ;

[0039] S306: Combine the discrete sampling matrix Z of the original measured surface shape with the fitted surface shape matrix. The pixel-by-pixel subtraction yields the surface shape result after Legendre polynomial filtering. .

[0040] Optionally, in step S305, the discrete values ​​of the fitted surface shape are... Reshape into a fitted surface matrix of the same size as the original measurement mesh. This includes the discrete values ​​of the fitted surface shape. The grid is rearranged into a two-dimensional matrix in a row-first, column-second, left-to-right, and top-to-bottom order to serve as the fitting surface matrix of the same size as the original measurement grid. .

[0041] Optionally, the discrete values ​​of the fitted surface shape The grid is rearranged into a two-dimensional matrix in a row-first, column-second, left-to-right, and top-to-bottom order to serve as the fitting surface matrix of the same size as the original measurement grid. When, let the fitted surface matrix The row and column indices are:

[0042] ;

[0043] ;

[0044] in, and These are the fitted surface shape matrices. row and column indexes, , Effective pixel count The discrete values ​​of the fitted surface shape are the row number of the discrete sampling matrix Z. and fitted surface matrix The mapping relationship is as follows:

[0045] ,

[0046] in, To fit the surface shape matrix, and These are the fitted surface shape matrices. row and column indexes, for In the k Fitted surface shape values ​​at each effective pixel.

[0047] Optionally, the step of concatenating the filtered shape results using a least squares algorithm to obtain the final measurement result includes:

[0048] S401: Establish a unified full-aperture coordinate system, with the overall geometric center of the X-ray plane mirror under test as the origin, and establish a global rectangular coordinate system (X,Y);

[0049] S402: Calculate the overlap error between sub-apertures for any pair of effectively overlapping sub-apertures. and Extract the filtered post-filter shape data within the overlapping region and construct the error vector. ;

[0050] S403: Use plane fitting to transform the error vector Dimensionality reduction to the error equation: ,in For sub-aperture and Error vector after plane fitting Let the coordinates of the point be... For sub-aperture and pitch error between For sub-aperture and Rolling error between For sub-aperture and Piston error between;

[0051] S404: Establish a global linear equation system to combine the error equations of all sub-aperture pairs, and the functional expression of the established global linear equation system is:

[0052] ;

[0053] in, Let be the vector of global correction parameters to be determined, where ~ The pitch error is for the first to m sub-apertures. ~ The rolling error is for the first to m sub-apertures. ~ For the piston error of the 1st to mth sub-diameters, the superscript indicates... Indicates the transpose operation; To construct the index matrix, rows correspond to overlapping regions, and columns correspond to unknown correction parameters. , The total number of effective overlapping areas. Number of sub-apertures; For all sub-apertures and Error vector after plane fitting The observation error vector of the overlapping region;

[0054] S405: Solve the global optimal correction parameters for the global linear equation system, including the global optimal pitch error, global optimal roll error, and global optimal piston error;

[0055] S406: Sub-aperture correction, including: mapping the obtained global optimal correction parameters back to each sub-aperture, performing pixel-by-pixel correction, and finally obtaining the aligned sub-aperture surface data in the global coordinate system;

[0056] S407: Weighted average stitching of all corrected sub-aperture data within the overlapping area to finally output the full-aperture surface shape measurement result.

[0057] Optionally, the function expression for mapping the obtained global optimal correction parameters back to each sub-aperture in step S406 is as follows:

[0058] ;

[0059] in, For sub-aperture exist Mapping results of interferometric measurements at the location, For sub-aperture exist Interferometric measurement results at the location, Let the coordinates of the point be... For sub-aperture The global optimal pitch error. For sub-aperture The globally optimal roll error, For sub-aperture The globally optimal piston error.

[0060] Compared with the prior art, the present invention can mainly achieve the following beneficial effects: (1) Legendre polynomials are a series of orthogonal polynomials with weighted functions in the interval [-1,1]. Their orthogonality ensures that any two Legendre polynomials of different orders do not affect each other in the rectangular domain, thus guaranteeing the uniqueness and stability of the fitting coefficients. They are particularly suitable for the decomposition and reconstruction of surface errors of rectangular optical elements. Their low-order terms (0, 1, 2) correspond to low-frequency aberrations such as translation, tilt, and defocus, while the high-order terms gradually introduce mid-to-high frequency spatial frequency components, with clear physical meaning. This invention utilizes Legendre polynomials to fit and analyze interferometric measurement data, which can accurately identify and separate measurement errors, mainly low-frequency ones, such as environmental disturbances, while retaining mid-frequency surface shape information that is crucial for slope error correction. This invention utilizes Legendre polynomial filtering to preemptively remove low-frequency measurement errors from sub-aperture data within the traditional least-squares splicing framework, significantly suppressing the cumulative effect of low-frequency errors in white light interferometers during splicing. This improves the mid-frequency surface shape error measurement accuracy of 200 mm-class X-ray plane mirrors from several nanometers PV in existing technologies to sub-nanometer PV, meeting the metrological requirements of post-processing after ion beam frequency division. (2) This invention does not require additional angle monitoring hardware; it can achieve sub-nanometer-level splicing accuracy solely through algorithms, simplifying system configuration and reducing equipment costs and assembly complexity. (3) This invention removes the first 10 Legendre low-frequency terms before splicing, resulting in splicing repeatability better than 0.1 nm PV and 180° flip self-consistent accuracy better than 0.3 nm PV, significantly improving long-term stability and data reliability. (4) The method of the present invention is compatible with existing white light interferometric splicing hardware. It can be adapted to measurement scenarios with different apertures and different overlap rates by simply adjusting the filter order. It has good versatility and engineering promotion value. Attached Figure Description

[0061] Figure 1 This is a basic flowchart of the method in an embodiment of the present invention.

[0062] Figure 2 This is a schematic diagram of the splicing system in an embodiment of the present invention, wherein 1 is a white light interferometer, 2 is a splicing motion platform, 3 is a stripe zeroing platform, and 4 is an automatic splicing control module.

[0063] Figure 3This is a detailed flowchart illustrating the process of obtaining a set of interferometric surface shape results with overlapping spatial positions of adjacent sub-apertures by white light interferometric splicing measurement of an X-ray plane mirror in an embodiment of the present invention.

[0064] Figure 4 This is a detailed flowchart illustrating how Legendre polynomials are used to fit the surface shape of each sub-aperture to obtain the polynomial coefficients of that surface shape in an embodiment of the present invention.

[0065] Figure 5 This is a detailed flowchart of obtaining the surface shape result of Legendre polynomial filtering in an embodiment of the present invention.

[0066] Figure 6 This is a detailed flowchart of how the filtered shape results are spliced ​​together using the least squares algorithm to obtain the final measurement result in an embodiment of the present invention.

[0067] Figure 7 The sub-aperture surface shape results obtained by measurement in the embodiments of the present invention are shown.

[0068] Figure 8 The sub-aperture surface shape obtained after Legendre polynomial filtering in the embodiments of the present invention is shown.

[0069] Figure 9 This is the final splicing measurement result obtained in the embodiment of the present invention. Detailed Implementation

[0070] To enable those skilled in the art to better understand the technical solutions of the present invention, the technical solutions of the present invention will be further described in detail below with reference to the accompanying drawings in the embodiments of the present invention.

[0071] like Figure 1 As shown, the mid-frequency surface shape white light interferometric stitching measurement method based on polynomial filtering in this embodiment includes the following steps: performing white light interferometric stitching measurement on an X-ray plane mirror to obtain a set of interferometric measurement surface shape results where adjacent sub-apertures have overlapping spatial positions; fitting the surface shape result of each sub-aperture using Legendre polynomials to obtain the preset order of the surface shape result. The polynomial coefficients are as follows; based on the preset order. The polynomial coefficients are fitted to generate corresponding order fitted surface shapes, and all order fitted surface shapes are subtracted from the measurement results to obtain a set of surface shape results after Legendre polynomial filtering. The filtered surface shape results are then stitched together using a least squares algorithm to obtain the final measurement result. This embodiment of the Legendre polynomial filtering method for measuring mid-frequency surface shape error using white-light interferometry stitching can utilize a white-light interferometer and a traditional least squares stitching algorithm to meet the measurement accuracy requirements of mid-frequency surface shape error of X-ray plane mirrors. This method improves measurement accuracy, reduces system costs, and enhances scalability. The preset order... The value can be selected according to actual needs. For example, as an optional implementation, the order is preset in this embodiment. .

[0072] Figure 2 This is a schematic diagram of the stitching system in this embodiment. The stitching system includes a white light interferometer 1, a stitching motion platform 2, a fringe zeroing platform 3, and an automatic stitching control module 4. The automatic stitching control module 4 is used to execute the mid-frequency surface shape white light interferometric stitching measurement method based on polynomial filtering in this embodiment. It includes a stitching motion control module, a fringe zeroing control module, a data measurement and acquisition module, and a filtering and stitching calculation module. The stitching motion control module is used to perform the displacement of all sub-aperture stitching; the fringe zeroing module performs zeroing fringe operation after the mirror under test moves to the new sub-aperture; the data measurement and acquisition module performs interferometric measurement to obtain the current sub-aperture surface shape data after the sub-aperture fringe is zeroed; the filtering and stitching calculation module performs Legendre polynomial fitting and filtering on the obtained surface shape data, and finally stitches the filtered sub-aperture data into full-aperture surface shape data.

[0073] like Figure 3 As shown, in this embodiment, white light interferometry splicing measurement is performed on the X-ray plane mirror to obtain a set of interferometric measurement surface shape results where the spatial positions of adjacent sub-apertures overlap, including:

[0074] S101: Based on the surface dimensions (Size) of the X-ray plane mirror being measured. x,y ) and the reference plane R of the white light interferometer microscope objective. x,y The entire surface area T to be measured is divided into m sub-apertures. The spatial positions of the m sub-apertures are arranged sequentially from left to right, and the spacing d between the sub-apertures satisfies a given overlap rate O. The m local sub-apertures can be represented as T i (i=1,2,…,m), in this embodiment, T i The spatial positions are arranged in order from left to right, and the sub-aperture spacing d satisfies the overlap rate O=40%;

[0075] S102: Position the X-ray plane mirror to the corresponding position on the stitching measurement system. Initially set the sub-aperture number i to 1. Drive the motion platform of the stitching system to move the mirror under test to the i-th sub-aperture T. i Location;

[0076] S103: The i-th sub-aperture T i Establish a local coordinate system L at the center i The origin is located at the i-th sub-aperture T. i The geometric center, Z i Along the i-th sub-aperture T i The normal arrangement drives the splicing system's electronically controlled two-dimensional adjustment table to adjust the number of interference fringes to zero;

[0077] S104: The piezoelectric ceramic of the white light interferometer drives the interference objective along the Z-axis. i The axial direction is scanned vertically based on a given scan step size s, and multiple interferograms are acquired in a single scan.

[0078] S105: The zero optical path difference point is extracted using a frequency domain algorithm to obtain the i-th sub-aperture T. i Interferometric measurement results M i , specifically Figure 7 As shown;

[0079] S106: Drive the motion platform of the stitching system to move the mirror under test to the (i+1)th sub-aperture T. i+1 The position is determined and the process jumps to step S103 until the surface shape results of all m sub-apertures are obtained.

[0080] like Figure 4 As shown, in this embodiment, the Legendre polynomial is used to fit the surface shape result of each sub-aperture to obtain the preset order of the surface shape result. When applying polynomial coefficients, the fitting of the surface shape results for each sub-aperture includes:

[0081] S201: Obtain the surface shape M to be analyzed i The discrete sampling matrix Z is used to record the effective pixel coordinates; in this embodiment, the size of the discrete sampling matrix Z is... The unit is millimeters;

[0082] S202: Establish the surface result M i Using a two-dimensional Cartesian coordinate system with the image center at the origin and pixel center spacing in units, generate unnormalized coordinate vectors. and ;

[0083] S203: Construction and Analysis of Surface Shape M i Mask matrix Z of the same size mask Mark all elements without a value (NaN) as invalid points, and only retain the valid pixels corresponding to the valid pixel coordinates;

[0084] S204: Transform the mask matrix Z mask The unnormalized coordinate vector corresponding to the effective pixels and Recombining the surface values ​​in the discrete sampling matrix Z into a one-dimensional vector, it can be expressed as:

[0085] ;

[0086] in, A one-dimensional vector representing the x-axis coordinate. ~ For the 1st to An unnormalized x-axis coordinate vector, A one-dimensional vector representing the y-axis coordinate. ~ For the 1st to An unnormalized y-axis coordinate vector A one-dimensional vector of surface shape values. ~ For the 1st to Each face value, and has ;

[0087] S205: Set the normalized width and height of the rectangular aperture, and map the surface matrix of the rectangular aperture to the normalized domain [-1,1]×[-1,1], thereby obtaining the normalized rectangular domain of the rectangular aperture;

[0088] S206: Construct a two-dimensional Legendre orthogonal polynomial basis matrix within a normalized rectangular domain. ;

[0089] S207: Based on the two-dimensional Legendre orthogonal polynomial basis matrix Constructing a least-squares linear system ,in Let the Legendre coefficient vector be... For effective surface shape magnitude vector, , ;

[0090] S208: Solving least-squares linear systems using one-dimensional vectors Obtain the preset order The surface shape M to be analyzed below i Legendre coefficient vector Used as the preset order for obtaining the surface shape result. The polynomial coefficients (in millimeters).

[0091] In step S202 of this embodiment, an unnormalized coordinate vector is generated. and satisfy:

[0092] ;

[0093] ;

[0094] in, and Let Z be the number of rows and columns of the discrete sampling matrix.

[0095] In step S206 of this embodiment, a two-dimensional Legendre orthogonal polynomial basis matrix is ​​constructed within the normalized rectangular domain. When constructing the two-dimensional Legendre orthogonal polynomial basis matrix The dimension is ,in Effective pixel count For the basis order, the basis order The value is set to the preset order. And the two-dimensional Legendre orthogonal polynomial basis matrix Any k-th column is given by the following formula:

[0096] ;

[0097] in, The basis matrix of the two-dimensional Legendre orthogonal polynomials For any k-th column, For about Two-dimensional substrate, They are respectively for Normalized coordinate vector, For about of A one-dimensional Legendre polynomial of order 1 For about of One-dimensional Legendre polynomial of order 1, order satisfy And under the same total order, according to Decreasing Arranged in ascending order, and having:

[0098] ;

[0099] in, and Let Z be the number of rows and columns of the discrete sampling matrix.

[0100] Furthermore, the one-dimensional Legendre polynomial is calculated using the Rodrigues expansion:

[0101] ;

[0102] in, For about and The one-dimensional Legendre polynomial. Regarding Two-dimensional substrate Through Cartesian product get.

[0103] like Figure 5 As shown, in this embodiment, according to the preset order The polynomial coefficients are fitted to generate the corresponding order of fitted surface shapes, and all order fitted surface shapes are subtracted from the measurement results to obtain a set of surface shape results after Legendre polynomial filtering, including:

[0104] S301: Input indicates the preset order Legendre coefficient vector of polynomial coefficients ,Right now:

[0105] ;

[0106] in, For the first The amplitude coefficient corresponding to the order base, in millimeters;

[0107] S302: Establish a two-dimensional grid coordinate matrix of the same size as the surface to be fitted and establish coordinate vectors at the pixel centers; specifically, in this embodiment, a two-dimensional grid coordinate matrix of the same size as the surface to be fitted is established. And establish coordinate vectors based on pixel centers ,in This is a vector transformation operation;

[0108] S303: Set a normalization factor to map pixel coordinates to a standardized rectangular area; specifically, in this embodiment, this involves setting a normalization factor. , Map pixel coordinates to a normalized rectangular area:

[0109] ;

[0110] S304: Construct a two-dimensional Legendre orthogonal polynomial basis matrix within a normalized rectangular domain. The two-dimensional Legendre orthogonal basis matrix in this embodiment , its first List by Given, among which satisfy And under the same total order, according to Decreasing Arranged in ascending order;

[0111] S305: Calculate the discrete values ​​of the fitted surface shape. ,in For the Legendre coefficient vector; the discrete values ​​of the fitted surface shape Reshape into a fitted surface matrix of the same size as the original measurement mesh. ;

[0112] S306: Discrete sampling matrix of the original measured surface shape Fitted surface matrix The pixel-by-pixel subtraction yields the surface shape result after Legendre polynomial filtering. ,like Figure 8 As shown.

[0113] In step S305 of this embodiment, the discrete values ​​of the fitted surface shape are... Reshape into a fitted surface matrix of the same size as the original measurement mesh. This includes the discrete values ​​of the fitted surface shape. The grid is rearranged into a two-dimensional matrix in a row-first, column-second, left-to-right, and top-to-bottom order to serve as the fitting surface matrix of the same size as the original measurement grid. In this embodiment, the discrete values ​​of the fitted surface shape will be used. The grid is rearranged into a two-dimensional matrix in a row-first, column-second, left-to-right, and top-to-bottom order to serve as the fitting surface matrix of the same size as the original measurement grid. When, let the fitted surface matrix The row and column indices are:

[0114] ;

[0115] ;

[0116] in, and These are the fitted surface shape matrices. row and column indexes, , Effective pixel count Discrete sampling matrix The number of rows, the discrete values ​​of the fitted surface shape and fitted surface matrix The mapping relationship is as follows: ,in To fit the surface shape matrix, and These are the fitted surface shape matrices. row and column indexes, for In the k Fitted surface shape values ​​at each effective pixel.

[0117] like Figure 6 As shown, in this embodiment, the filtered shape results are concatenated using the least squares algorithm to obtain the final measurement result, including:

[0118] S401: Establish a unified full-aperture coordinate system, with the overall geometric center of the X-ray plane mirror under test as the origin, and establish a global rectangular coordinate system. ,in Along the long axis of the mirror, Along the minor axis of the mirror; each sub-aperture in its local coordinate system The overlapping area between adjacent sub-apertures is defined as follows: ; For sub-aperture and The overlapping area between them This is the intersection operator;

[0119] S402: Calculate the overlap error between sub-apertures for any pair of effectively overlapping sub-apertures. and Extract the filtered post-filter shape data within the overlapping region and construct the error vector. ,Right now: ;

[0120] S403: Use plane fitting to transform the error vector Dimensionality reduction to the error equation: ,in For sub-aperture and Error vector after plane fitting Let the coordinates of the point be... For sub-aperture and pitch error between For sub-aperture and Rolling error between For sub-aperture and Piston error between;

[0121] S404: Establish a global linear equation system to combine the error equations of all sub-aperture pairs, and the functional expression of the established global linear equation system is:

[0122] ;

[0123] in, Let be the vector of global correction parameters to be determined, where ~ The pitch error is for the first to m sub-apertures. ~ The rolling error is for the first to m sub-apertures. ~ For the piston error of the 1st to mth sub-diameters, the superscript indicates... Indicates the transpose operation; To construct the index matrix, rows correspond to overlapping regions, and columns correspond to unknown correction parameters. , The total number of effective overlapping areas. Number of sub-apertures; For all sub-apertures and Error vector after plane fitting The observation error vector of the overlapping region is as follows:

[0124] ,

[0125] in, For sub-aperture and Error vector after plane fitting For sub-aperture and Error vector after plane fitting For sub-aperture and The error vector after plane fitting;

[0126] S405: Solve for the globally optimal correction parameters of the global linear equation system, including the globally optimal pitch error, globally optimal roll error, and globally optimal piston error. This can be calculated using singular value decomposition to compute the Moore-Penrose pseudoinverse.

[0127] ;

[0128] in, The globally optimal correction parameters include each sub-aperture. Global optimal pitch error Global optimal roll error and the global optimal piston error ;

[0129] S406: Sub-aperture correction, including: mapping the obtained global optimal correction parameters back to each sub-aperture, performing pixel-by-pixel correction, and finally obtaining the aligned sub-aperture surface data in the global coordinate system;

[0130] S407: Weighted average stitching of all corrected sub-aperture data within the overlapping area to finally output the full-aperture surface shape measurement result.

[0131] In step S406 of this embodiment, the function expression for mapping the obtained global optimal correction parameters back to each sub-aperture is as follows:

[0132] ;

[0133] in, For sub-aperture exist Mapping results of interferometric measurements at the location, For sub-aperture exist Interferometric measurement results at the location, Let the coordinates of the point be... For sub-aperture The global optimal pitch error. For sub-aperture The globally optimal roll error, For sub-aperture The globally optimal piston error.

[0134] In step S407 of this embodiment, the function expression for weighted average stitching of all corrected sub-aperture data within the overlapping region is as follows:

[0135] ;

[0136] in, The full-aperture surface shape measurement results are obtained after global optimization and least-squares stitching, such as... Figure 9 As shown. For sub-aperture exist Mapping results of interferometric measurements at the location, For sub-aperture exist Weight at the location, and sub-aperture The total weight of all positions is 1.

[0137] In summary, the method in this embodiment effectively removes low-frequency errors in white light interferometer measurement results through Legendre polynomial filtering, and achieves high-precision mid-frequency surface shape measurement by combining it with a global optimization least squares stitching algorithm. This solves the problem of insufficient accuracy caused by the accumulation of low-frequency errors in traditional white light interferometric stitching technology. This method is simple to operate, requires no additional hardware support, and significantly reduces system complexity and cost. Compared with existing technologies, this method not only overcomes the accuracy limitations of traditional methods but also achieves broad adaptability to different measurement scenarios by flexibly adjusting the filtering order. It provides an efficient and accurate surface shape measurement solution for the optical manufacturing and inspection fields, and has significant industrial application value and prospects.

[0138] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A mid-frequency surface shape white light interferometric splicing measurement method based on polynomial filtering, characterized in that, The process includes the following steps: performing white-light interferometry on the X-ray plane mirror to obtain a set of interferometric surface shape results where adjacent sub-apertures have overlapping spatial positions; fitting the surface shape result of each sub-aperture using Legendre polynomials to obtain the preset order of the surface shape result. The polynomial coefficients are as follows; based on the preset order. The polynomial coefficients under the given conditions are used to generate the corresponding order of fitted surface shapes, and all order fitted surface shapes are subtracted from the measurement results to obtain a set of surface shape results after Legendre polynomial filtering. The filtered shape results are then concatenated using the least squares algorithm to obtain the final measurement result, including: S401: Establish a unified full-aperture coordinate system, with the overall geometric center of the X-ray plane mirror under test as the origin, and establish a global rectangular coordinate system (X,Y); S402: Calculate the overlap error between sub-apertures for any pair of effectively overlapping sub-apertures. and Extract the filtered post-filter shape data within the overlapping region and construct the error vector. ; S403: Use plane fitting to transform the error vector Dimensionality reduction to the error equation: ,in For sub-aperture and Error vector after plane fitting Let be the coordinates of the point. For sub-aperture and pitch error between For sub-aperture and Rolling error between For sub-aperture and Piston error between; S404: Establish a global linear equation system to combine the error equations of all sub-aperture pairs, and the functional expression of the established global linear equation system is: ； in, Let be the vector of global correction parameters to be determined, where ~ The pitch error is for the first to m sub-apertures. ~ The rolling error is for the first to m sub-apertures. ~ For the piston error of the 1st to mth sub-diameters, the superscript indicates... Indicates the transpose operation; To construct the index matrix, rows correspond to overlapping regions, and columns correspond to unknown correction parameters. , The total number of effective overlapping areas. Number of sub-apertures; For all sub-apertures and Error vector after plane fitting The observation error vector of the overlapping region; S405: Solve the global optimal correction parameters for the global linear equation system, including the global optimal pitch error, global optimal roll error, and global optimal piston error; S406: Sub-aperture correction, including: mapping the obtained global optimal correction parameters back to each sub-aperture, performing pixel-by-pixel correction, and finally obtaining the aligned sub-aperture surface data in the global coordinate system; S407: Weighted average stitching of all corrected sub-aperture data within the overlapping area to finally output the full-aperture surface shape measurement result.

2. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 1, characterized in that, The method of performing white-light interferometry stitching measurements on the X-ray plane mirror to obtain a set of interferometric surface shape results with overlapping spatial positions of adjacent sub-apertures includes: S101: Based on the surface dimensions (Size) of the X-ray plane mirror being measured. x,y ) and the reference plane R of the white light interferometer microscope objective. x,y The entire surface T to be measured is divided into m sub-apertures, and the spatial positions of the m sub-apertures are arranged sequentially from left to right, with the sub-aperture spacing d satisfying the given overlap rate O. S102: Position the X-ray plane mirror to the corresponding position on the stitching measurement system. Initially set the sub-aperture number i to 1. Drive the motion platform of the stitching system to move the mirror under test to the i-th sub-aperture T. i Location; S103: The i-th sub-aperture T i Establish a local coordinate system L at the center i The origin is located at the i-th sub-aperture T. i The geometric center, Z i Along the i-th sub-aperture T i The normal arrangement drives the splicing system's electronically controlled two-dimensional adjustment table to adjust the number of interference fringes to zero; S104: The piezoelectric ceramic of the white light interferometer drives the interference objective along the Z-axis. i The axial direction is scanned vertically based on a given scan step size s, and multiple interferograms are acquired in a single scan; S105: The zero optical path difference point is extracted using a frequency domain algorithm to obtain the i-th sub-aperture T. i Interferometric measurement results M i ; S106: Drive the motion platform of the stitching system to move the mirror under test to the (i+1)th sub-aperture T. i+1 The position is determined and the process jumps to step S103 until the surface shape results of all m sub-apertures are obtained.

3. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 1, characterized in that, The surface shape of each sub-aperture is fitted using Legendre polynomials to obtain a preset order for the surface shape result. When applying polynomial coefficients, the fitting of the surface shape results for each sub-aperture includes: S201: Obtain the surface shape M to be analyzed i The discrete sampling matrix Z is used to record the effective pixel coordinates; S202: Establish the surface result M i Using a two-dimensional Cartesian coordinate system with the image center at the origin and pixel center spacing in units, an unnormalized coordinate vector is generated. and ; S203: Construction and Analysis of Surface Shape M i Mask matrix Z of the same size mask Mark all elements without a value (NaN) as invalid points, and only retain the valid pixels corresponding to the valid pixel coordinates; S204: Transform the mask matrix Z mask The unnormalized coordinate vector corresponding to the effective pixels and The surface values ​​in the discrete sampling matrix Z are recombined into one-dimensional vectors; S205: Set the normalized width and height of the rectangular aperture, and map the surface matrix of the rectangular aperture to the normalized domain [-1,1]×[-1,1], thereby obtaining the normalized rectangular domain of the rectangular aperture; S206: Construct a two-dimensional Legendre orthogonal polynomial basis matrix within a normalized rectangular domain. ; S207: Based on the two-dimensional Legendre orthogonal polynomial basis matrix Constructing a least-squares linear system ,in Let the Legendre coefficient vector be... The effective surface shape magnitude vector; S208: Solving least-squares linear systems using one-dimensional vectors Obtain the preset order The surface shape M to be analyzed below i The Legendre coefficient vector is used as the preset order for obtaining the surface shape result. The polynomial coefficients under the given conditions.

4. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 3, characterized in that, The size of the discrete sampling matrix Z in step S201 is In step S202, an unnormalized coordinate vector is generated. and satisfy: ； ； in, and Let Z be the number of rows and columns of the discrete sampling matrix.

5. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 4, characterized in that, In step S206, a two-dimensional Legendre orthogonal polynomial basis matrix is ​​constructed within the normalized rectangular domain. When constructing the two-dimensional Legendre orthogonal polynomial basis matrix The dimension is Where N is the number of effective pixels, The base order is the number of bases. The value is set to the preset order. And the two-dimensional Legendre orthogonal polynomial basis matrix Any k-th column is given by the following formula: ； in, The basis matrix of the two-dimensional Legendre orthogonal polynomials For any k-th column, For about Two-dimensional substrate, They are respectively for Normalized coordinate vector, For about of A one-dimensional Legendre polynomial of order 1 For about of One-dimensional Legendre polynomial of order 1, order satisfy And under the same total order, according to Decreasing Arranged in ascending order, and having: ； in, and Let Z be the number of rows and columns of the discrete sampling matrix.

6. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 1, characterized in that, The according to the preset order The polynomial coefficients are fitted to generate the corresponding order of fitted surface shapes, and all order fitted surface shapes are subtracted from the measurement results to obtain a set of surface shape results after Legendre polynomial filtering, including: S301: Input indicates the preset order Legendre coefficient vector of polynomial coefficients ; S302: Establish a two-dimensional grid coordinate matrix with the same size as the surface to be fitted and establish coordinate vectors with pixel centers; S303: Set the normalization factor to map pixel coordinates to a normalized rectangular area; S304: Construct a two-dimensional Legendre orthogonal polynomial basis matrix within a normalized rectangular domain. ; S305: Calculate the discrete values ​​of the fitted surface shape. ,in For the Legendre coefficient vector; the discrete values ​​of the fitted surface shape Reshape into a fitted surface matrix of the same size as the original measurement mesh. ; S306: Combine the discrete sampling matrix Z of the original measured surface shape with the fitted surface shape matrix. The pixel-by-pixel subtraction yields the surface shape result after Legendre polynomial filtering. .

7. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 6, characterized in that, In step S305, the discrete values ​​of the fitted surface shape are... Reshape into a fitted surface matrix of the same size as the original measurement mesh. This includes the discrete values ​​of the fitted surface shape. The grid is rearranged into a two-dimensional matrix in a row-first, column-second, left-to-right, and top-to-bottom order to serve as the fitting surface matrix of the same size as the original measurement grid. .

8. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 7, characterized in that, The discrete values ​​of the fitted surface shape The grid is rearranged into a two-dimensional matrix in a row-first, column-second, left-to-right, and top-to-bottom order to serve as the fitting surface matrix of the same size as the original measurement grid. When, let the fitted surface matrix The row and column indices are: ； ； in, and These are the fitted surface shape matrices. row and column indexes, , Effective pixel count The discrete values ​​of the fitted surface shape are the row number of the discrete sampling matrix Z. and fitted surface matrix The mapping relationship is as follows: ， in, To fit the surface shape matrix, and These are the fitted surface shape matrices. row and column indexes, for In the k Fitted surface shape values ​​at each effective pixel.

9. The method for mid-frequency surface shape white light interferometric splicing measurement based on polynomial filtering according to claim 1, characterized in that, In step S406, the function expression for mapping the obtained global optimal correction parameters back to each sub-aperture is as follows: ； in, For sub-aperture exist Mapping results of interferometric measurements at the location, For sub-aperture exist Interferometric measurement results at the location, Let be the coordinates of the point. For sub-aperture The global optimal pitch error. For sub-aperture The globally optimal roll error, For sub-aperture The globally optimal piston error.

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