An encoding method for computing holography

Abstract

This application proposes an encoding method for holographic computation. The method includes: classifying different types of curves based on the morphological distribution of etched lines; calculating the coordinates of discrete points on the etched lines; dividing the index intervals of different curves according to their feature points using Newton's iteration method; and fitting the arc by finding the optimal solution of the set of feature point coordinates that can be fitted using a bisection method. The curves are closed curves, spiral curves, or non-spiral curves. This encoding method significantly reduces the amount of data required for etched line coordinates, achieving encoding with a small data volume, and the calculation process is simple. High-precision encoding can be achieved by setting the encoding precision.

Description

Technical Field

[0001] This application belongs to the field of optical technology and relates to an encoding method for computing holography. Background Technology

[0002] With the development of optical precision processing technology, the application of aspherical surfaces is becoming increasingly widespread. The deviation of aspherical surfaces is approximately tens to hundreds of micrometers, and the current trend indicates that it will continue to increase. Therefore, high-precision inspection of aspherical surfaces with high steepness and large deviation is a major challenge in the field of optical inspection.

[0003] Computational holography is one of the effective methods for achieving high-precision detection of aspherical surfaces. The computational holographic element (CGH) can convert the standard spherical wave emitted by the interferometer into an ideal wavefront that matches the surface to be measured, thus achieving zero-position detection. Therefore, it can achieve high-precision detection of various curved surfaces.

[0004] Computational holography is a diffractive optical element that relies on micro-nano lithography technology. The processing equipment can only read a digital curve composed of a finite number of feature coordinate points. However, the amount of etched line coordinate data generated by numerical encoding of the compensated phase is enormous, severely impacting the data reading speed of the lithography machine and limiting the widespread application of high-precision computational holography. Therefore, utilizing as few data points as possible to reduce the read / write load on the processing equipment is crucial for high-precision detection of computational holography.

[0005] To address the massive amount of data related to the coordinates of the scribing lines, existing techniques employ a variable-step-size search encoding method to calculate the coordinates of the scribing lines and then use a straight-line fitting method for description. While this method reduces the amount of data to some extent, the data volume remains large, and the calculation process is complex. Summary of the Invention

[0006] To overcome the above-mentioned shortcomings, the purpose of this application is to propose an encoding method that reduces the amount of data for the scribe line coordinates while meeting the encoding accuracy requirements.

[0007] To achieve the above-mentioned objectives, this application adopts the following technical solution:

[0008] An encoding method for computing holograms, characterized in that the method comprises:

[0009] Based on the morphological distribution of the scribing lines, different types of curves are classified, and the coordinates of the discrete points of the scribing lines are calculated.

[0010] Based on Newton's iteration method, different curves are divided into index intervals according to their feature points. The optimal solution for fitting the arc is found by using the bisection method to find the set of feature point coordinates that can be fitted to the arc.

[0011] The curve can be a closed curve, a spiral curve, or a non-spiral curve.

[0012] In one embodiment, calculating the coordinates of the discrete points of the scribe lines includes:

[0013] Based on the CGH surface phase function Φ(x,y), a nonlinear function f(x,y,N)=Φ(x,y)-N is established with respect to the coordinates (x,y) and the fringe number N;

[0014] Numerical solution to the nonlinear equation f(x, y, N) = 0,

[0015] By using contour lines to obtain the coordinates of a subset of discrete points of the etched stripes, and combining this with the specific shape of the etched stripes, the starting and ending point coordinates and the index interval in the x-direction can be determined.

[0016] Calculate the y value corresponding to each x value using Newton's iteration method.

[0017] In one embodiment, the encoding method for computing holography includes,

[0018] Set the step size and iteration precision, starting from the initial point.

[0019] Using calculation formula Calculate the y value corresponding to each x value.

[0020] Where Y0 is the initial value of the iteration, y p The y-value is calculated for the p-th coordinate iteration. Let be the partial derivative of the nonlinear function of the p-th coordinate with respect to y.

[0021] In one embodiment, the coordinates x-values ​​of the first and last points of the scribed stripe are x. start x end ,as well as

[0022] The coordinates of the two points corresponding to the minimum and maximum values ​​in the x-direction are x. min x max ,

[0023] The closed curve is characterized by x. start =x end ,

[0024] The characteristic of the cyclotron curve is x start ≠x end And x start ≠x min ,

[0025] The non-rotation curve is characterized by x start ≠x end And x start =x min .

[0026] In one embodiment, the selection of the initial value y0 for the iteration depends on the shape of the scribing stripes.

[0027] The closed curve and the spiral curve are in the interval [x] start x min When ), the initial value of the iteration is Y0 = y start ;

[0028] The closed curve and the spiral curve are in the interval [x] min x max When Y0 = 0, the initial value of the iteration is 0.

[0029] The closed curve and the spiral curve are in the interval [x] min x end When ), the initial value of the iteration is Y0 = y end ;

[0030] The non-rotation curve is in the interval [x] start x end When Y0 = 0, the initial value of the iteration is 0.

[0031] In one embodiment, the circular arc fitting includes:

[0032] The center coordinates (x0, y0) and radius R0 of the fitted circular arc for the n discrete points can be obtained by calculation.

[0033] Calculate the starting and ending angles (θ1, θ2) of the arc based on the coordinates of the two points at the beginning and end of the arc.

[0034] In one embodiment, the encoding method for computing holography includes:

[0035] Calculate the fitting accuracy based on the arc parameters; based on the relationship between the fitting accuracy and the accuracy setting value eps, the bisection method steps are as follows:

[0036] Add or remove 0.5Δ coordinates from the original fitted coordinate interval, W n =W n-1 ±int(0.5 n Δ),

[0037] Among them, W n-1 W represents the size of the coordinate interval in the previous loop. n Let n be the size of the coordinate interval in the next loop, n be the number of loops to find the optimal solution for this arc, and Δ be the set number of coordinates to add or remove.

[0038] The coordinate interval to be fitted is continuously updated until the optimal solution of the coordinate set is found.

[0039] In one embodiment, in the encoding method for computing holography,

[0040] All discrete points obtained by encoding the etched stripes are used to describe the arc curve, thereby obtaining a fitting curve of the segmented arc that meets the fitting accuracy requirements.

[0041] S1. Numerical coding

[0042] The coordinates of the discrete points of the scribing stripes were calculated using Newton's iteration method.

[0043] Based on the phase function Φ(x, y), a nonlinear function is established with respect to the coordinates (x, y) and the fringe ordinal number N.

[0044] f(x,y,N)=Φ(x,y)-Nπ#(1)

[0045] The search process is transformed into a numerical solution process for the nonlinear equation (2).

[0046] f(x, y, N) = 0#(2)

[0047] The coordinates of a subset of discrete points of the etched stripe are obtained using contour lines. Combined with the specific shape of the stripe, the start and end point coordinates and the index interval in the x-direction are determined. Then, the step size and iteration precision are set, and starting from the initial point, the y-value corresponding to each x-value is calculated using Newton's iteration method.

[0048] The formula used in Newton's iteration method is as follows:

[0049]

[0050] Where Y0 is the initial value of the iteration, y p The y-value is calculated for the p-th coordinate iteration. Let be the partial derivative of the nonlinear function of the p-th coordinate with respect to y.

[0051] S2. Curve Description

[0052] The circular arc fitting method and the bisection method are used to obtain the circular arc fitting curve of the scribing stripes.

[0053] The general expression for a circle is,

[0054]

[0055] Where x and y are the coordinates of discrete points, x0 and y0 are the coordinates of the center of the circle, and R0 is the radius of the circle.

[0056] For each point, substituting into equation (4) yields a linear equation. For n discrete points, the system of linear equations is:

[0057]

[0058] in,

[0059]

[0060] Convert equation (5) into matrix form,

[0061] AX = B#(7)

[0062] in,

[0063]

[0064] Calculate the least squares solution of equation (7).

[0065] X = (A T A) -1 A T B#(9)

[0066] Based on X, we can obtain the parameters of the fitted circular arc for these n discrete points: the center coordinates (x0, y0) and the radius R0.

[0067] Furthermore, based on the coordinates of the two points at the beginning and end of this arc, the starting and ending angles (θ1, θ2) of this arc can be calculated.

[0068] The above is the basic calculation process for circular arc fitting of discrete coordinates. Next, we will find the optimal solution of the set of discrete coordinates that can meet the fitting accuracy requirements.

[0069] Based on the arc parameters obtained in the previous step, calculate the fitting accuracy. Using the bisection method, and considering the relationship between the arc fitting accuracy and the accuracy setting value eps, add or remove 0.5Δ coordinates from the original fitted coordinate interval.

[0070] W n =W n-1 ±int(0.5 n Δ)#(10)

[0071] Among them, W n-1 W represents the size of the coordinate interval in the previous loop. n Δ represents the size of the coordinate interval in the next iteration, n represents the number of iterations to find the optimal solution for this arc, and Δ represents the set number of coordinates to add or remove.

[0072] The coordinate range to be fitted is continuously updated, and then the fitting calculation is performed again until the optimal solution of the coordinate set is found.

[0073] Beneficial effects

[0074] Compared with the prior art, the coding method proposed in this application has the following advantages:

[0075] 1) It greatly reduces the amount of data for the scale line coordinates, enabling small data readings and simplifying the calculation process.

[0076] 2) High-precision encoding can be achieved by setting the encoding precision.

[0077] 3) Increasing the data processing speed of the lithography machine and reducing machine time will help reduce costs. Attached Figure Description

[0078] The accompanying drawings, which are included in and form part of this specification, are used together with the embodiments of this application to explain the principles of this application, but do not constitute a limitation thereof.

[0079] Figure 1 This is a schematic diagram of the CGH compensation detection optical path in this application.

[0080] Figure 2 This is a flowchart of Newton's iteration method in numerical encoding, which is an exemplary embodiment of this application.

[0081] Figure 3 This is a schematic diagram illustrating the classification of etched stripe patterns in an exemplary embodiment of this application.

[0082] Figure 4 This is a schematic diagram of a circular arc fitting curve for discrete points according to an exemplary embodiment of this application.

[0083] Figure 5 This is a schematic diagram of the fitting curve error for an exemplary embodiment of this application.

[0084] Figure 6 This is a flowchart illustrating the determination of the optimal solution for circular arc fitting based on the bisection method in an embodiment of this application.

[0085] Figure 7 This is a scribed stripe pattern shown according to an exemplary embodiment of this application.

[0086] Figure 8 This is a schematic diagram of a circular arc fitting curve according to an exemplary embodiment of this application. Detailed Implementation

[0087] This application proposes an encoding method for holographic computation. The method includes: classifying different types of curves based on the morphological distribution of etched lines; calculating the coordinates of discrete points on the etched lines; dividing the index intervals of different curves according to their feature points using Newton's iteration method; and fitting the arc by finding the optimal solution of the set of feature point coordinates that can be fitted using a bisection method. The curves are closed curves, spiral curves, or non-spiral curves. This encoding method significantly reduces the amount of data required for etched line coordinates, achieving encoding with a small data volume, and the calculation process is simple. High-precision encoding can be achieved by setting the encoding precision.

[0088] Next, the encoding method for computing holograms proposed in this application will be described with reference to the accompanying drawings.

[0089] Figure 1 This is a schematic diagram of the optical path for CGH compensation detection. The optical path consists of an interferometer, a CGH, and the surface under test. The etched surface of the CGH is closer to the interferometer, and the +1st order diffracted light is used as the theoretical detection wavefront. The distance from the interferometer focal point F to the CGH is d1, and the distance from the CGH to the center point of the surface under test is d2. The spherical wave emitted by the interferometer is converted into a wavefront matching the surface shape of the surface under test after passing through the CGH. After reflection from the surface of the surface under test, it returns to the interferometer along the original optical path, completing the interferometric detection. The surface shape distribution of the surface under test is obtained based on the detection results.

[0090] Figure 2 This is a flowchart of Newton's iteration method in numerical coding. The calculation process is as follows: Based on the phase distribution of the CGH surface, a nonlinear function is established with respect to coordinates x, y, and fringe ordinal number N, transforming the search process into a numerical solution process for the nonlinear equation f(x, y, N) = 0. Within the index interval, the y value corresponding to each x value is calculated using Newton's iteration method.

[0091] Figure 3 This is a schematic diagram illustrating the classification of scribe line morphology. Since the calculation results of Newton's iteration method depend on the selection of initial values, the results are based on the coordinates x-values ​​of the first and last points of the scribe lines. start x end And the coordinates of the two points corresponding to the minimum and maximum values ​​in the x-direction. min x max The stripes are classified into: closed curves, spiral curves, and non-spiral curves.

[0092] The specific division principles and index ranges are shown in Table 1.

[0093] Table 1. Principles and Index Ranges for Stripe Classification

[0094]

[0095] According to the division principle, stripes with the same stripe contour type are merged, and iterative calculations are performed in different index intervals between the start and end points of the stripes.

[0096] Correspondingly, the initial value Y0 for iteration varies for different index intervals, as shown in Table 2. Where y start y end The coordinates y-values ​​of the first and last points of the stripe.

[0097] Table 2 Initial values ​​for iteration in different intervals

[0098]

[0099] Figure 4 This is a schematic diagram of the circular arc fitting curve for discrete points in Embodiment 1 of this application. The curve is composed of multiple circular arcs of different colors, that is, these segmented circular arcs represent the original curve. It can be seen that where the curvature changes significantly, a larger number of segmented circular arcs are required; where the curvature changes slightly, a smaller number of segmented circular arcs are required.

[0100] Figure 5 This diagram illustrates the error of the fitted curve. The solid line represents the ideal curve, and the dashed lines represent the upper and lower boundaries of the normal error δ. The error at any point is the shortest distance from that point to the circle, i.e.,

[0101]

[0102] The set of coordinates of the feature points that can be fitted to each arc segment is calculated based on the magnitude of the error.

[0103] Figure 6 This is a flowchart illustrating the process of determining the optimal solution for circular arc fitting based on the bisection method. In the computational method of the circular arc fitting algorithm, the coordinate interval to be fitted is continuously updated based on the bisection method, searching for the optimal solution of the set of feature point coordinates that satisfies the fitting accuracy requirements of the circular arc.

[0104] Next, the above solution will be further described with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of protection of this application. The implementation conditions used in the embodiments can be further adjusted as in specific experiments, and the implementation conditions not specified are generally those in conventional experiments.

[0105] Example 1:

[0106] Simulations are performed using discrete coordinate points of the sine function y = 5sinx, such as... Figure 4 As shown, the curve is composed of multiple arcs of different colors; these segmented arcs represent the original curve. It can be seen that where the curvature changes significantly, a larger number of segmented arcs are needed; where the curvature changes slightly, a smaller number of segmented arcs are needed.

[0107] Example 2:

[0108] The parameters of the aspherical surface to be tested for CGH compensation testing are shown in Table 3. It is an aluminum parabolic reflector with a radius of curvature of 500 mm and a diameter of 220 mm. The system structural parameters are shown in Table 4.

[0109] Table 3 Aspherical Parameters

[0110]

[0111] Table 4 System Structure Parameters

[0112]

[0113] Ray tracing is used to obtain the compensated phase distribution of light rays passing through the CGH surface, that is, phase data is used for encoding calculation.

[0114] Assume the phase function is Φ(x, y) and the nonlinear function is f(x, y, N).

[0115] Based on the morphological classification principle of scribed stripes, stripes with the same overall outline type are merged. Then, their index range is determined according to their feature points; and iterative calculations are performed in different index intervals.

[0116] SI. Numerical coding

[0117] Taking a 500π phase etched line as an example, such as Figure 7 As shown, its expression is,

[0118] f(x,y,500)=Φ(x,y)-500π=0#(12)

[0119] Using contour lines, we obtain the coordinates of some discrete points of this stripe, and determine the start and end point coordinates as: (-17.3504, 8.1327), (17.3504, 8.1327). According to Table 1, this stripe is a non-spiral curve, and its index interval is: [-17.3504, 17.3504].

[0120] As shown in Table 2, the initial value Y0 for the iteration of this index interval is 0.

[0121] Then, the step size is set to 0.01 mm and the iteration accuracy is 1e-7.

[0122] After determining the coefficients, starting from the initial point, use Newton's iteration method to calculate the y value corresponding to each x value.

[0123]

[0124] This yields a series of discrete point coordinates for the etched stripe, thus completing the numerical encoding calculation.

[0125] S2. Curve Description

[0126] The discrete points obtained from the numerical encoding are described by curves. Let the fitting accuracy setpoint eps be 5e-5, the number of coordinates added or removed Δ be 30, and the initial coordinate interval W0 also be 30.

[0127] Fit the coordinate range to be fitted to obtain the curve parameters and fitting accuracy.

[0128] Based on the relationship between the fitting accuracy and the accuracy setting value eps, and using the bisection method, add or remove 0.5Δ coordinates from the original fitted coordinate interval to determine whether the fitting accuracy meets the requirements.

[0129] Suppose that the fitting accuracy of the first calculation does not meet the requirements, then,

[0130] W1 = W0 - iht(0.5) 1 Δ)=15#(14)

[0131] Where W1 is the coordinate interval of the second cycle.

[0132] Assuming the fitting accuracy of the first calculation meets the requirements, then,

[0133] W1 = W0 + int(0.5) 1 Δ)=45#(15)

[0134] Then update the coordinate interval to be fitted, and perform the fitting calculation again until the optimal solution of the coordinate set is found, that is, the best-fit arc.

[0135] This yields the parameters for the best-fit arc: the center coordinates (x0, y0) and the radius R0. Furthermore, based on the coordinates of the first and last two points of this arc, the starting and ending angles (θ1, θ2) of the arc are calculated.

[0136] Calculate the stripe and draw the arc, as follows: Figure 8 As shown.

[0137] After numerical encoding and curve description of all the scribing lines on the CGH surface, the numerical encoding error using circular arc fitting is approximately 0.006λ. Compared to the traditional linear fitting description method, the encoding method proposed in this application reduces the amount of data by approximately 82% while improving fitting accuracy.

[0138] Table 5 Comparison of data between linear and curvilinear descriptions

[0139]

[0140] By drawing the graphic based on the parameters of all the arcs, the curve description of the scribing stripes is completed.

[0141] The above embodiments are only for illustrating the technical concept and features of this application, and are intended to enable anyone skilled in the art to understand the content of this application and implement it accordingly. They should not be construed as limiting the scope of protection of this application. All equivalent changes or modifications made in accordance with the spirit and essence of this application should be included within the scope of protection of this application.

Claims

1. An encoding method for computing holograms, characterized in that, The method comprises: According to the distribution of the shape of the ruled line, different types of curves are divided, and the discrete point coordinates of the ruled line are calculated, Based on Newton iteration method, the index interval is divided according to the characteristic points of different curves, The optimal solution of the characteristic point coordinate set that can be fitted by the circular arc is found by dichotomy to fit the circular arc, The curve is a closed curve or a spiral curve or a non-spiral curve; The calculation of the discrete point coordinates of the ruled line comprises: According to the CGH surface phase function Φ (x, y), a nonlinear function f (x, y, N) = Φ (x, y) - N about coordinates (x, y) and stripe order N is established; Numerical solution of the nonlinear equation f (x, y, N) = 0 is obtained, The discrete point coordinates of a part of the ruled line are obtained by using the contour line, and the start and end point coordinates and the index interval in the x direction are determined by combining the specific shape of the ruled line, The y value corresponding to each x value is calculated by using Newton iteration method, The step size and iteration accuracy are set, and the iteration starts from the initial point, Using the calculation Calculate the y value corresponding to each x value, where Y0is an initial value of iteration, y p is the y value of the pth coordinate iteration calculation, is the partial derivative of the pth coordinate nonlinear function to y, The coordinate x value of the first and last points of the ruled line stripe is x start , x end , and the coordinate of the two points corresponding to the minimum and maximum values in the x direction is x min , x max , The closed curve feature is x start = x end , The clothoid feature is x start ≠ x end and x start ≠ x min , The non-helical curve feature is x start ≠ x end and x start = x min , The selection of the iteration initial value Y0 depends on the shape of the ruled line, The closed curve and the evolute curve are in the interval [x start , x min ), the iteration initial value Y0=y start ; The closed curve and the evolute curve are in the interval [x min , x max ), the iteration initial value Y0=0. The closed curve and the evolute curve are in the interval [x min , x end ), the iteration initial value Y0=y end ; The non-helical curve is in the interval [x start , x end ], and the iteration initial value Y0=0.

2. The encoding method for calculating hologram according to claim 1, characterized in that, The fitting of the circular arc comprises: The center coordinates (x0, y0) and the radius R0 of the fitting circular arc of n discrete points can be obtained by calculation; according to the coordinates of the first and last points of the circular arc, the start and end angles (θ1, θ2) of the circular arc are calculated.

3. The encoding method for calculating hologram according to claim 2, characterized in that, According to the circular arc parameters, the fitting accuracy is calculated; according to the size relationship between the fitting accuracy of the circular arc and the accuracy setting value eps, the dichotomy step is: W = W ± int(0.5 Δ) in the coordinate interval of the original fit n = W n-1 ± int(0.5 n Δ), where W n-1 is the size of the coordinate interval in the previous cycle, W n is the size of the coordinate interval in the next cycle, n is the number of cycles for finding the optimal solution for the circular arc, and Δ is the number of coordinates to be added or removed; the coordinate interval to be fitted is constantly updated until the optimal solution of the coordinate set is found.

Images