CN118152709A - Dwell time algorithm based on wavelet basis fitting - Google Patents

Abstract

The invention relates to the technical field of optical processing residence time calculation, in particular to a residence time algorithm based on wavelet base fitting. Comprising the following steps: measuring the surface shape of a workpiece to be processed through an interferometer to obtain a surface shape residual error; selecting a processing technology and performing fixed-point processing measurement to obtain a removal function; setting a polishing track type and polishing track parameters according to the surface shape sampling size of a workpiece to be processed; expressing the surface shape of the workpiece to be processed by using wavelet to obtain a surface shape spectrum; expressing a removal function by using a wavelet to obtain a removal function feature matrix; and constructing a matrix equation by using the surface shape spectrum and the removal function feature matrix, and solving the residence time spectrum by using an iteration method so as to calculate the residence time. The advantages are that: the advantage of local fitting of wavelets is utilized, so that the residence time solution is more accurate in a small scale range, thereby realizing accurate removal, and being beneficial to the accurate control of the residence time of elements with extremely high roughness requirements, such as a strong laser reflector or a photoetching objective lens; has repeatability.

Description

Dwell time algorithm based on wavelet basis fitting

Technical Field

The invention relates to the technical field of optical processing dwell time calculation, and in particular to a dwell time algorithm based on wavelet basis fitting.

Background technique

The multi-resolution analysis method is a method of performing image analysis at different resolutions in the field of imaging, but its theory can be applied to characterize surface shape residuals. The expression for solving the dwell time problem is as follows:

;

in, represents convolution, /> Represents the trajectory coverage.

At present, there are mainly five methods for dwell time calculation: the first is the Bayesian method, which is mainly introduced in the paper "Algorithm for ion beam figuring of low-gradient mirrors"; the second is the Fourier transform method, which is mainly introduced in the paper "Iterative blind deconvolution method for dwell-time adjustment"; the third is the matrix method, which is mainly introduced in the paper "Algorithm for dwell time solution of magnetorheological processing of large-aperture optical components"; the fourth is the direct convolution method, which is mainly introduced in the paper "Dwell-time algorithm for polishing large optics".

The above four methods each have their own advantages, but they all have a problem, that is, obtaining the dwell time through iterative correction of the solution, which will result in a non-smooth solution and put the dynamic performance of the machine tool to the test.

The fifth method is the polynomial fitting method. This method fits the solution through a smooth analytical polynomial, and the solution obtained will avoid the above-mentioned machine tool dynamic performance problems. The paper "Zernike mapping of optimum dwell time indeterministic fabrication of freeform optics" mainly introduces this method. However, this method is a global fitting and cannot correct the local solution, so that the dwell time solution will deviate from the solution of the original problem. Therefore, the solution of the existing dwell time algorithm still cannot balance the dynamic performance of the machine tool and the local precise dwell time well.

Summary of the invention

In order to solve the above problems, the present invention provides a dwell time algorithm based on wavelet basis fitting.

The present invention aims to provide a dwell time algorithm based on wavelet basis fitting, which specifically comprises the following steps:

S1. Measure the surface shape of the workpiece to be processed by interferometer to obtain the discretized surface shape residual of the workpiece to be processed ;

S2. Select the processing technology of the workpiece to be processed and measure the removal function ;

S3. Set the polishing trajectory type and polishing trajectory parameters according to the surface sampling size of the workpiece to be processed, and the dwell point on the trajectory is ;

S4. Using wavelet to express the surface shape of the workpiece to be processed: Select Gaussian scaling function and/> , construct a one-dimensional scaling function family and a one-dimensional wavelet function family:

;

In the formula, represents the scale factor, /> represents the displacement coefficient;

Using the one-dimensional scaling function family and the one-dimensional wavelet function family, a two-dimensional wavelet function family is constructed according to the tensor construction method in multi-resolution analysis. 、/> 、/> 、/> ;

The residual error of the workpiece surface to be processed Expressed as:

;

In the formula, It is a face shape/> and/> The number of discretized points in the direction. The wavelet translation Limited to face size Within, /> is the scale factor, /> is the 2D vertical detail coefficient, /> is the 2D horizontal detail coefficient, /> is the two-dimensional oblique detail coefficient. Two-dimensional scaling function/> , two-dimensional wavelet functions are divided into three categories, the first category is vertical wavelet/> , the second type is horizontal wavelet , the third type is oblique wavelet/> . /> They correspond to the horizontal, vertical and inclined numbers respectively;/> is the maximum detail scale number, /> is the minimum detail scale number;

Rewritten into column vector form:

;

Where:

The wavelet basis vectors are:

The surface spectrum is:

;

S5. Use wavelet to express the removal function and transform the convolution kernel into the two-dimensional Taylor expansion:

;

In the formula, Is the removal function/> The wavelet feature matrix of

S6. Solve the fitting coefficient matrix equation of the construction equation: ; In the formula, /> is the dwell time spectrum;

right The calculation accuracy is estimated and the residual is set as:

;

The error equation vector is:

;

in, Yes/> The standard inner product of is the error measure vector for the two-dimensional approximate scale space,/> is the error metric vector for the two-dimensional vertical, horizontal and oblique wavelet space,/> is the total error metric vector of the calculation method;

Pair Regularize and use iterative method to solve C ;

S7. Calculate the final residence time :/> .

Preferably, step S2 is as follows: select a test piece of the same material as the workpiece to be processed, measure its initial surface shape, and then measure the surface shape again after polishing for 10 seconds, subtract the surface shapes measured before and after processing, and then divide by the polishing time 10 seconds to obtain the removal function within 1 second. .

Preferably, the iteration target set in the iteration method of step S6 is:

;

in, is the minimum dwell time due to the dynamic performance limitations of the machine tool,/> is the minimum step length on the trajectory, It is the maximum speed limit of the machine tool; /> is the regularization factor. /> It means to find the 2-norm of a matrix or vector;

The iterative objective is solved to obtain a solution C.

Preferably, the method for solving the iterative target in step S6 is Newton's method, gradient descent method, conjugate gradient method, Krylov subspace approximation method, biorthogonalization method or SBB method.

Compared with the prior art, the present invention can achieve the following beneficial effects:

The solution method of the present invention utilizes the advantages of local fitting of wavelets, making the dwell time solution more accurate within a small scale range, which is conducive to the precise control of the dwell time of components with extremely high roughness requirements such as strong laser reflectors or photolithography lenses. The method is also repeatable and can calculate the precise dwell time for the local area, thereby achieving precise removal and guiding the processing practice of components such as strong laser reflectors or photolithography lenses.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a residence time algorithm based on wavelet basis fitting provided according to the present invention.

FIG. 2 is a core workflow program framework diagram provided according to the present invention.

Detailed ways

Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, the same modules are represented by the same reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, the detailed description thereof will not be repeated.

In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation of the present invention.

The dwell time algorithm based on wavelet basis fitting specifically includes the following steps:

S1. Measure the surface shape of the workpiece to be processed by interferometer to obtain the discretized surface shape residual of the workpiece to be processed ;

S2. Select the processing technology of the workpiece to be processed and measure the removal function ;

S3. Set the polishing trajectory type and polishing trajectory parameters according to the surface sampling size of the workpiece to be processed, and the dwell point on the trajectory is ;

S4. Use wavelet to express the surface shape of the workpiece to be processed: select Gaussian scaling function and/> , construct a one-dimensional scaling function family and a one-dimensional wavelet function family:

(1)

Construct two-dimensional wavelet based on tensor construction method in multi-resolution analysis; construct two-dimensional scaling function by tensor product , two-dimensional wavelet functions are divided into three categories. The first category is vertical wavelet , the second type is horizontal wavelet/> , the third type is oblique wavelet/> .

The tensor construction method expression is as follows:

(2)

in, Represents the tensor product. The construction rule of is:/> (3)

According to the multi-resolution analysis theory formula , where /> It is expressed as a direct sum. It is known that the residual error of the workpiece surface to be processed/> can be expressed as:

(4)

Make some restrictions on the expression to facilitate calculation; is the maximum detail scale number, /> is the minimum detail scale number; for example, the Gaussian wavelet support length is selected as/> , and the surface size is the diameter of , the field size for observing roughness is/> ;

but ;

;

Among them, ceil (*) is rounded up, and floor (*) is rounded down. Indicates the support set of the function. In the formula, /> It is a face shape/> and/> The number of discretized points in the direction. The wavelet translation Limited to the surface size/> Within, /> is the scale factor, /> is the 2D vertical detail coefficient, /> is the 2D horizontal detail coefficient, /> is the two-dimensional oblique detail coefficient. /> They correspond to the horizontal, vertical and inclined numbers respectively.

Then rewrite (4) into column vector inner product form:

(5)

in:

The wavelet basis vectors are:

(6)

The surface spectrum is:

(7)

S5. Use wavelet to express the removal function and transform the convolution kernel into the two-dimensional Taylor expansion:

(8)

Is the removal function/> The wavelet feature matrix; this step can refer to the paper "Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet"

S6. Solve the fitting coefficient matrix equation of the construction equation:

;

in, is the dwell time spectrum;

This equation is based on the following considerations:

A known:

So:

Due to the orthogonality of the wavelet basis:

So we get:

So we get the equation .

Estimate the calculation accuracy of the above formula and set the residual as:

;

The error equation vector is:

;

in, Yes/> The standard inner product of is the error measure vector for the two-dimensional approximate scale space,/> is the error metric vector for the two-dimensional vertical, horizontal and oblique wavelet space,/> is the total error metric vector of the calculation method;

The problem represented is still an ill-posed problem, so Tikhonov regularization is needed to improve stability and it is solved by iteration. The iteration goal is set as:

(16)

in, is the minimum dwell time due to the dynamic performance limitations of the machine tool,/> is the minimum step length on the trajectory, It is the maximum speed limit of the machine tool. /> is the regularization factor. /> It means to find the 2-norm of a matrix or vector.

right The formula is regularized and C is solved by iterative method.

The method for solving the iterative target is Newton's method, gradient descent method, conjugate gradient method, Krylov subspace approximation method, biorthogonalization method or SBB method.

S7. Calculate the final residence time :/> ; The solution is complete.

The residence time solution flow chart proposed in this patent is shown in Figure 1, and the code framework is shown in Figure 2. Indicates obtaining corresponding data, /> Indicates setting the corresponding data, ceil (*) is rounded up, floor (*) is rounded down, Indicates the support set of the function. /> is the initial value of the iteration, /> Indicates the function of seeking/> The independent variable is at its minimum value The value of .

The core of the present invention is: select Gaussian scaling function and wavelet function for data preprocessing. The solution is made smoother by fitting method to meet the dynamic performance of machine tools. By taking advantage of the local fitting of wavelets, the dwell time solution is made more accurate within a small scale range, which is conducive to the precise control of the dwell time of components with extremely high roughness requirements such as strong laser reflectors or photolithography objective lenses. The method is repeatable and can calculate the precise dwell time for local areas, thereby achieving precise removal and guiding the processing practice of components such as strong laser reflectors or photolithography objective lenses.

The method proposed in the present invention belongs to the fifth polynomial fitting method described in the background technology, but it relies on the theoretical background of multi-resolution analysis to realize local correction of the solution and analyze high-frequency information, thereby accurately controlling the roughness of the processed component. This makes the solution more accurate in terms of residence time while meeting the dynamic performance problem of the machine tool.

In summary, the advantage of the present invention is that it can effectively solve the influence of the algorithm on the roughness in the calculation of the residence time of components with relatively high local roughness requirements. The core idea is to apply the powerful ability of wavelets in small-scale correction to solve the residence time problem. While ensuring the smoothness of the solution, the accurate residence time can also be obtained, which can provide an effective means for controlling local roughness.

It should be understood that the various forms of processes shown above can be used to reorder, add or delete steps. For example, the steps described in the disclosure of the present invention can be performed in parallel, sequentially or in different orders, as long as the desired results of the technical solution disclosed in the present invention can be achieved, and this document does not limit this.

The above specific implementations do not constitute a limitation on the protection scope of the present invention. It should be understood by those skilled in the art that various modifications, combinations, sub-combinations and substitutions can be made according to design requirements and other factors. Any modification, equivalent substitution and improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (4)

1. The residence time algorithm based on wavelet base fitting is characterized by comprising the following steps:

S1, measuring the surface shape of a workpiece to be processed through an interferometer to obtain a discretized surface shape residual error of the workpiece to be processed ；

S2, selecting a processing technology of the workpiece to be processed, and measuring a removal function；

S3, setting polishing track types and polishing track parameters according to the surface shape sampling size of the workpiece to be processed, wherein resident points on the tracks are as follows；

S4, expressing the surface shape of the workpiece to be processed by using wavelet: selecting a Gaussian scale functionAnd/>Constructing a one-dimensional scale function family and a one-dimensional wavelet function family:

；

In the method, in the process of the invention, Representing scale factors,/>Representing a displacement coefficient;

constructing a two-dimensional wavelet function family according to a tensor construction method in multi-resolution analysis by utilizing a one-dimensional scale function family and a one-dimensional wavelet function family 、/>、/>、/>；

Residual error of the surface shape of the workpiece to be processedExpressed as:

；

In the method, in the process of the invention, Is the shape of face/>And/>The number of directional discretized points. Shift wavelet amount/>Define in-plane shape size/>Within,/>Is the scale factor,/>Is a two-dimensional vertical detail coefficient,/>Is a two-dimensional horizontal detail coefficient,/>Is a two-dimensional oblique detail coefficient. Two-dimensional scale function/>The two-dimensional wavelet functions are divided into three classes, the first being vertical wavelets/>The second class is horizontal wavelet/>The third class is the oblique wavelet/>。/>Corresponding to horizontal, vertical and inclined numbers respectively; Is the maximum detail scale sequence number,/> Is the minimum detail scale number;

The rewritten column vector form is as follows:

；

Wherein:

The wavelet basis vectors are:

The surface spectrum is:

；

S5, using wavelet to express a removal function, and converting the convolution kernel into the convolution kernel according to two-dimensional Taylor expansion:

；

In the method, in the process of the invention, Is a removal function/>Is a wavelet feature matrix of (a);

S6, solving a fitting coefficient matrix equation of the construction equation: ; in the/> Is the residence time spectrum;

For a pair of The calculation accuracy of (2) is estimated, and a residual error is set as follows:

；

The error equation vector is:

；

Wherein, Is/>Standard inner product of/>Is an error metric vector for a two-dimensional approximation scale space,Is an error metric vector for two-dimensional vertical, horizontal and diagonal wavelet spaces,/>Is the total error metric vector of the calculation method;

Opposite type Regularizing and solving the C by an iteration method;

S7, calculating final residence time ：/>。

2. The residence time algorithm based on wavelet base fitting according to claim 1, wherein said step S2 is specifically as follows: selecting an experimental piece made of the same material as the workpiece to be processed, measuring the initial surface shape, measuring the surface shape again after polishing time is 10s, subtracting the surface shapes measured before and after processing, and dividing the surface shape by the polishing time for 10s to obtain a removal function within 1s。

3. The residence time algorithm based on wavelet base fitting according to claim 2, wherein the iteration targets established in the step S6 iteration method are:

；

Wherein, Due to the minimum residence time of the machine tool dynamic performance limit,/>Is the minimum step distance on the track,/>Is the maximum speed limit of machine tool operation; /(I)Is a regularization factor. /(I)Representing a 2-norm of the matrix or vector;

and solving the iteration target to obtain a solution C.

4. A dwell time algorithm based on wavelet base fitting according to claim 3, wherein the method of solving the iterative objective in step S6 is newton' S method, gradient descent method, conjugate gradient method, krylov subspace approximation method, biorthogonal method or SBB method.