Design method of complex surface CGH gauge based on spline modeling

Abstract

This invention relates to the field of precision inspection technology, and more particularly to a method for designing complex surface CGH fixtures based on spline modeling. The method involves determining sampling points within the effective aperture range of the CGH, and defining the target phase for each sampling point. A B-spline function is used as the mathematical representation of the target phase to construct the current node model. The current node model is then optimized using the sampling points and their target phases to obtain the optimal node model and the corresponding optimal node matrix. Based on the optimal node matrix, data for CGH encoding and manufacturing is output. This invention solves the problems of geometric mismatch and poor local adaptability in fitting complex surfaces using traditional global basis functions through an error-driven hierarchical node generation strategy, achieving sub-picometer level design accuracy with a minimal number of parameters.

Description

Technical Field

[0001] This invention belongs to the field of precision testing technology, and in particular relates to a design method for complex surface CGH fixtures based on spline modeling. Background Technology

[0002] In modern optical manufacturing, the application of complex surfaces such as off-axis aspherical surfaces, cylindrical mirrors, and freeform surfaces is becoming increasingly widespread. The Computer Generated Hologram (CGH), as a diffractive optical element capable of generating arbitrarily complex wavefronts, is the core compensator for achieving high-precision zero-position detection of such surfaces. The design goal of the CGH is to convert the standard reference wave (such as a plane wave or spherical wave) output by the interferometer into a target wavefront that perfectly matches the theoretical shape of the measured surface.

[0003] Existing CGH design methods mainly employ global polynomial fitting (such as Zernike polynomials and XY polynomials) or fixed grid sampling (Grid Phase). When dealing with complex surface shapes such as large apertures, large off-axis measurements, or non-rotationally symmetric surfaces (such as cylindrical surfaces), existing technologies suffer from the following significant drawbacks:

[0004] (1) Geometric mismatch problem: Zernike polynomials are defined on a unit circle and have rotational symmetry. However, the apertures of many complex optical elements are elongated or rectangular, and their surfaces exhibit unidirectional curvature. Forcibly fitting the cylindrical phase of a rectangular region with the Zernike polynomial of a circular region will lead to a serious mathematical mismatch;

[0005] (2) Global basis function fitting is prone to oscillation or local underfitting: When the target phase changes drastically in a local area (such as off-axis, freeform surface edge, local steep region), the fixed-order global polynomial / fixed basis function may show local residual increase or oscillation phenomenon, which requires increasing the order, leading to model instability and parameter expansion.

[0006] (3) The contradiction between data volume and accuracy: In order to achieve nanometer-level detection accuracy, the traditional direct sampling method requires an extremely high sampling rate, resulting in a large data file and difficulty in optimization;

[0007] (4) Poor local adaptability: Complex surfaces often have drastic local curvature changes (such as areas with large off-axis amounts), while other areas are flat. Traditional global polynomial or uniform sampling cannot allocate computing resources as needed according to local features, which can easily lead to undersampling or oversampling, which is not conducive to subsequent manufacturing file generation, spatial frequency analysis and error control. Summary of the Invention

[0008] In view of this, the present invention aims to provide a design method for CGH fixtures with complex surfaces based on spline modeling. It constructs a spline phase model through an error-driven hierarchical node adaptive refinement strategy, and outputs phase data that can be used for CGH coding, manufacturing and inspection. This solves the problems of geometric mismatch and poor local adaptability of traditional global basis functions in fitting complex surfaces, and achieves sub-picometer level design accuracy with very few parameters.

[0009] To achieve the above objectives, the technical solution created by this invention is implemented as follows:

[0010] A method for designing CGH fixtures for complex surfaces based on spline modeling, comprising:

[0011] S1: Determine the sampling points within the effective aperture range of the CGH, and determine the target phase for each sampling point;

[0012] S2: Using B-spline functions as the mathematical representation of the target phase, construct the current node model; using the sampling points and their target phase obtained in step S1, optimize the current node model to obtain the optimal node model and the optimal node matrix corresponding to the optimal node model.

[0013] S3: Based on the optimal node matrix obtained in step S2, output the data used for CGH encoding manufacturing.

[0014] Furthermore, in step S1: the target phase of each sampling point is determined by the following formula:

[0015] ;

[0016] in, i Let λ represent the target phase at the i-th sampling point, and λ represent the wavelength of the reference light emitted to the CGH. OPD i This represents the optical path difference between the reference light and the target light at the i-th sampling point.

[0017] Furthermore, the process of constructing the current node model in step S2 includes:

[0018] S21: From all the sampling points in the selection step S1, uniformly select a small number of sampling points as nodes, and construct the initial spline function of the nodes with phase as output and sampling point coordinates as input;

[0019] S22: Calculate the difference between the initial spline function in step S21 and the target phase corresponding to all sampling points to obtain the initial global residual distribution mapped to each node interval;

[0020] S23: Based on the residual distribution within each node interval obtained in step S22, find the interval to be optimized, and dynamically insert new nodes into all intervals to be optimized;

[0021] S24: Merge the new node inserted in step S23 with the original node, maintaining the node density unchanged; construct the current spline function using the obtained node according to step S21.

[0022] S25: Based on the current spline function obtained in step S24, construct an overdetermined system of equations with all target phases as objectives and spline coefficients as unknowns;

[0023] S26: Solve the minimum norm solution of the overdetermined system of equations in step S25, obtain the spline coefficients with the minimum global residual sum of squares under the current node distribution, and obtain the current node model. The corresponding current node distribution is the current node matrix.

[0024] Furthermore, the process of dynamically inserting new nodes into all intervals to be optimized in step S23 includes:

[0025] S231: Sort all intervals to be optimized in descending order of their maximum residual values ​​to obtain a sorted index sequence;

[0026] S232: From the intervals sorted in step S231, select the first η% intervals to be optimized as the set to be encrypted in this round, and insert a new node at the geometric midpoint of the η% intervals to be optimized to obtain multiple new node intervals;

[0027] S233: Calculate the interval width of each new node interval in step S232, and repeat the operation of steps S231 to S232 for all new node intervals whose interval width is greater than or equal to the preset minimum node spacing threshold.

[0028] Furthermore, the overdetermined system of equations in step S25 is as follows:

[0029] ;

[0030] Where J(c) represents the overdetermined system of equations for solving the spline coefficients c of the spline function, μ k This represents the surface shape of the k-th sampling point. target (μ k ) represents the target phase corresponding to the k-th sampling point, M represents the total number of sampling points, and c i Let N represent the i-th spline coefficient in the spline function, and let n represent the total number of spline coefficients. i,p Let represent the p-th term B-spline basis function in the spline function.

[0031] Furthermore, the stopping condition optimized in step S2 is that the maximum residual of the currently obtained node matrix is ​​less than a preset residual threshold.

[0032] Furthermore, step S2 also includes: using the analytical properties of B-splines and numerical iterative algorithms to verify the obtained optimal node matrix.

[0033] Furthermore, the verification process includes: determining multiple sampling points mutually exclusive with those in step S1 within the effective aperture range of the CGH, and constructing a verification point set; based on the obtained optimal node model, directly and analytically calculating the phase gradient at any verification point within the effective aperture of the CGH using the derivative property of the B-spline; determining the inverse vector of the light emitted from the CGH based on the calculated phase gradient; and establishing the surface ray parameter equation from the CGH plane to the optical element under test based on the inverse vector, thus obtaining the emission point of the light emitted from the CGH.

[0034] ;

[0035] in, This represents the inverse vector, and t represents the optical path parameter. This represents the spatial coordinates of the ray corresponding to the optical path parameter t. This represents the initial spatial coordinates of the ray on the CGH plane, i.e., the coordinates of the ray's exit point on the CGH plane;

[0036] The CGH emits light rays from the emission point onto the surface of the optical element under test, and determines the normal vector at the intersection of the surface of the optical element under test and the light ray. Based on the normal vector, a positive ray path is constructed to return from the surface of the optical element under test to the CGH plane, and the return point of the returning ray and the CGH plane is obtained. The geometric position deviation and the ray angle deviation between the emission point and the return point are calculated.

[0037] Furthermore, the optical path parameter is obtained by iteratively solving the following equation:

[0038] ;

[0039] Among them, t j Let represent the optical path parameter obtained in the j-th iteration, F represent the surface shape equation of the optical element under test, and α represent the under-relaxation factor.

[0040] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0041] (1) The present invention provides a method for designing complex surface CGH fixtures based on spline modeling. This method abandons traditional global polynomial fitting or uniform grid sampling and adopts B-spline functions as the mathematical representation of the CGH phase, using least squares spline approximation instead of strict interpolation. By solving the overdetermined equations under a limited number of adaptive nodes, the spline curves are fitted to the massive ray data in the global scope with the best root mean square (RMS) fit. This not only achieves significant data compression but also utilizes the smoothing properties of the least squares method to naturally filter numerical noise, directly generating a smooth phase surface suitable for ultra-precision machining. The initial function is then compared with all sampling points in the original phase dataset to calculate the initial global fitting residual distribution, which serves as the basis for subsequent optimization.

[0042] (2) The complex surface CGH fixture design method based on spline modeling described in this invention obtains the final phase function through error-driven hierarchical adaptive iterative optimization: First, it automatically identifies local node intervals where the residual exceeds the preset accuracy threshold; then, it dynamically inserts new nodes only in the identified error-exceeding intervals for densification, while keeping the nodes sparse in the smooth regions where the residual meets the requirements, thereby realizing on-demand allocation of computing resources; finally, it updates the node matrix and recalculates the spline coefficients, and iteratively executes the above residual calculation, interval identification and node insertion steps, refining layer by layer until the global maximum fitting residual meets the preset design accuracy requirements, thereby obtaining the final CGH phase distribution function. Attached Figure Description

[0043] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments and descriptions of the invention are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0044] Figure 1 A flowchart illustrating the design method for complex surface CGH fixtures based on spline modeling as described in the embodiments of the present invention;

[0045] Figure 2 A schematic diagram of the geometric layout of the optical element under test - CGH - detection system described in an embodiment of the present invention;

[0046] Figure 3 A schematic diagram illustrating the process of constructing the current node model as described in an embodiment of the present invention;

[0047] Figure 4 The distribution diagram of macroscopic phase reconstruction described in the embodiments of the present invention;

[0048] Figure 5 The distribution diagram of the fitting residual space described in the embodiment of the present invention;

[0049] Figure 6 A graph illustrating the node refinement and convergence process described in the embodiments of the present invention.

[0050] Explanation of reference numerals in the attached figures:

[0051] 1. Interferometer; 2. Optical element under test; 3. CGH. Detailed Implementation

[0052] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof. It should be noted that, unless otherwise specified, the embodiments and features described herein can be combined with each other.

[0053] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on this invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, features defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0054] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0055] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0056] like Figure 1 As shown in the embodiment of the present invention, the method for designing complex surface CGH fixtures based on spline modeling includes:

[0057] S1: Determine the sampling points within the effective aperture range of the CGH, and determine the target phase for each sampling point.

[0058] In this embodiment of the invention, based on the theoretical equation z(x,y) of the optical element under test (which can be an off-axis aspherical surface, a freeform polynomial, an aspherical cylindrical surface, etc.) and the geometric layout of the detection system, an optical path difference / phase calculation model is established to determine the "CGH plane point - target phase". The model can be implemented using forward tracing or reverse tracing, and both are equivalent. The geometric layout of the optical element under test-CGH-detection system constructed in this embodiment of the invention is as follows: Figure 2 As shown, the detection system is an interferometer 1. Specifically, this invention uses an off-axis aspherical optical element as the optical element under test 2. That is, the interferometer 1 emits light to illuminate the CGH3, the light beam generated by the CGH3 reaches the optical element under test 2, and the light reflected by the optical element under test 2 returns to the interferometer 1 via the CGH3. In addition, in this embodiment of the invention, the sampling interval is limited to meet the coverage requirements of the target phase spatial frequency (e.g., less than the Nyquist interval of the highest spatial frequency of the target).

[0059] In some embodiments, the target phase of each sampling point is determined by the following formula:

[0060] ;

[0061] in, i Let λ represent the target phase at the i-th sampling point, λ represent the wavelength of the reference light emitted by interferometer 1 to CGH3, and OPD represent the wavelength of the reference light emitted by interferometer 1 to CGH3. i This represents the optical path difference between the reference light and the target light at the i-th sampling point.

[0062] S2: Using B-spline functions as the mathematical representation of the target phase, construct the current node model; using the sampling points and their target phase obtained in step S1, optimize the current node model to obtain the optimal node model and the optimal node matrix corresponding to the optimal node model.

[0063] In some embodiments, the process of constructing the current node model is as follows: Figure 3 As shown, it includes:

[0064] S21: From all the sampling points in selection step S1, uniformly select a small number of sampling points as nodes, and construct the initial spline function of the nodes with phase as output and sampling point coordinates as input. In this embodiment of the invention, the initial spline function is:

[0065] ;

[0066] Where, μ k Indicates the surface shape at the sampling point. Indicates phase, c i Let N represent the i-th spline coefficient in the spline function to be solved, n represent the total number of spline coefficients, and N represent the total number of spline coefficients. i,p Let represent the p-th term B-spline basis function in the spline function.

[0067] S22: Calculate the difference between the initial spline function in step S21 and the target phase corresponding to all sampling points to obtain the initial global residual distribution mapped to each node interval. In this embodiment of the invention, the initial global residual distribution is:

[0068] ;

[0069] Among them, e k This represents the residual at the k-th sampling point. target (μ k ) represents the k-th sampling point μ k The corresponding target phase.

[0070] S23: Based on the residual distribution within each node interval obtained in step S22, find the interval to be optimized, and dynamically insert new nodes into all intervals to be optimized.

[0071] This invention employs a selective node encryption strategy for inserting new nodes. Specifically, in some embodiments, the process of dynamically inserting new nodes into all intervals to be optimized includes:

[0072] S231: Sort all intervals to be optimized in descending order of their maximum residual value to obtain the sorted index sequence.

[0073] S232: From the intervals sorted in step S231, select the first η% of intervals to be optimized as the set to be encrypted in this round, and insert a new node at the geometric midpoint of the η% intervals to be optimized, resulting in multiple new node intervals. The node insertion probability η is a preset parameter; in this embodiment, the node insertion probability η is set to a typical value of 30 (i.e., selecting the first 30% of intervals to be optimized as the set to be encrypted in this round).

[0074] S233: Calculate the interval width of each new node interval in step S232, and repeat steps S231 to S232 for all new node intervals whose interval width is greater than or equal to the preset minimum node spacing threshold. This step can avoid numerical ill-conditioned problems caused by excessive node density.

[0075] S24: Merge the new node inserted in step S23 with the original node, maintaining the node density unchanged; construct the current spline function using the resulting nodes according to step S21. In some other embodiments, node merging methods include matrix merging, sorting, and deduplication. For smooth regions where the residuals already meet the accuracy requirements, the original node density remains unchanged, thereby achieving on-demand allocation of computing resources.

[0076] S25: Based on the current spline function obtained in step S24, construct an overdetermined system of equations with all target phases as objectives and spline coefficients as unknowns:

[0077] ;

[0078] Where J(c) represents the overdetermined system of equations for solving the spline coefficients c of the spline function, and M represents the total number of sampling points.

[0079] S26: Solve for the minimum norm solution of the overdetermined system of equations in step S25 to obtain the spline coefficients with the minimum global residual sum of squares under the current node distribution. This yields the current node model, and the corresponding current node distribution is the current node matrix. This step ensures that while locally refining the nodes, the overall model still maintains optimal smoothness.

[0080] In some embodiments, the optimized stopping condition in step S2 is that the maximum residual of the currently obtained node matrix is ​​less than a preset residual threshold. In other embodiments, the optimized stopping condition also includes reaching a preset maximum iteration level or node number limit, or the convergence rate being lower than a threshold.

[0081] To verify the rigorous correctness of the generated mathematical model in physical optics, this invention also uses a round-trip ray tracing model of "CGH—surface under test—CGH" (e.g., Figure 2 As shown, the optimal node matrix is ​​verified.

[0082] In some embodiments, step S2 further includes verifying the obtained optimal node matrix using the analytical properties of B-splines and a numerical iterative algorithm. The verification process includes:

[0083] Within the effective aperture range of the CGH, multiple sampling points mutually exclusive with those in step S1 are determined to construct a validation point set. In this embodiment of the invention, the validation point set is obtained using methods such as odd-even alternating sampling and random sampling, based on the positions of the sampling points in step S1. The validation point set is used to evaluate the generalization error of the optimal node model.

[0084] Based on the obtained optimal node model, the phase gradient at any verification point within the effective aperture of the CGH is directly calculated analytically using the derivative property of B-splines, as shown in the following equation:

[0085] ;

[0086] in, Represents the phase gradient. This represents the derivative of the B-spline basis function. This invention utilizes analytical derivatives to replace the difference approximation, eliminating truncation errors and ensuring the theoretical accuracy of the calculated angle of the outgoing ray.

[0087] Based on the calculated phase gradient, the reverse vector of the light rays emitted from the CGH is determined. Based on the reverse vector, the surface ray parameter equation from the CGH plane to the optical element under test is established, and the exit point of the light rays emitted from the CGH is obtained.

[0088] ;

[0089] in, This represents the inverse vector, and t represents the optical path parameter. This represents the spatial coordinates of the ray corresponding to the optical path parameter t. This represents the starting spatial coordinates of the ray on the CGH plane, i.e., the coordinates of the ray's exit point on the CGH plane.

[0090] According to the generalized Snell's law and the theory of local grating diffraction, the phase gradient on the CGH surface is equivalent to a transverse spatial frequency modulation applied to the incident wavefront. Based on the phase gradient at the verification point obtained through analytical calculation, the reverse unit direction vector of the light rays diffracted out from the CGH plane can be determined. In the embodiments of the present invention, such as... Figure 2 As shown, interferometer 1 emits light to CGH3. CGH3 receives the light from interferometer 1 and then emits light to the surface of the optical element 2 under test.

[0091] Since the surface equations of the optical element under test are usually high-order aspherical or freeform surface equations, the intersection points of light rays and the surface cannot be directly solved analytically. Therefore, this invention employs a Newton-Raphson iterative method with an under-relaxation factor to solve for the optical path parameter t. Specifically, the optical path parameter is obtained iteratively using the following formula:

[0092] ;

[0093] Among them, t j Let represent the optical path parameter obtained in the j-th iteration, F represent the surface shape equation of the optical element under test, and α represent the under-relaxation factor, which is used to prevent iterative divergence in steep surface regions and ensure that it can converge quickly and with high precision to the intersection of the surface of the optical element under test and the outgoing rays of CGH.

[0094] The CGH emits light rays from the emission point onto the surface of the optical element under test, and determines the normal vector at the intersection of the surface of the optical element under test and the light rays. According to the zero-position detection principle, it is assumed that the light rays are reflected (or return along the normal direction). At this time, based on the normal vector, a positive light ray path is constructed from the surface of the optical element under test back to the CGH plane, and the return point of the returning light rays and the CGH plane is obtained. The geometric position deviation and the light ray angle deviation between the emission point and the return point are calculated.

[0095] S3: Based on the optimal node matrix obtained in step S2, output the data used for CGH encoding manufacturing.

[0096] In this embodiment of the invention, the final node matrix and spline segment coefficients (pp-form or equivalent form) are exported, and data for CGH encoding manufacturing / simulation verification is output, including: uniform resampling phase table, derivative / spatial frequency report, and stripe period or phase encoding file for layout / writing.

[0097] The convergence of the fitting residuals before and after node encryption provided by this invention is as follows: Figures 4 to 6 As shown, the approximation accuracy, generalization ability, and iteration efficiency of the spline model described in this invention are objectively demonstrated. Specifically, Figure 4 The figure shows a comparison between the target phase within the effective aperture of the CGH and the fitting of the spline model. The cubic B-spline curve in the figure completely overlaps with the massive training data points, and the spline nodes generated in the final product achieve an adaptive non-uniform distribution driven by error in the computational domain. This result shows that low-order piecewise polynomials are sufficient to accurately characterize the nonlinear target phase with a large dynamic range, avoiding the numerical instability problem easily caused by global high-order polynomials from the bottom layer. Figure 5 This figure demonstrates the spatial distribution and generalization ability of the fitted residuals, revealing the microscopic residual state of the optimal nodal model across the entire aperture range. After adaptive optimization, the root mean square of the fitted residuals for both the training and independent validation sets converges to extremely low numerical levels. The residual scatter points exhibit a random and uniform distribution around the zero axis, without systematic bias or Runge phenomenon at aperture edges or high curvature regions. The validation set error is strictly consistent with the training set error, proving that the model possesses extremely high physical fidelity and generalization ability, completely eliminating the risk of overfitting. Figure 6 The optimization process of dynamically inserting new nodes based on the maximum residual is recorded, where the horizontal axis represents the refinement iteration level and the vertical axis represents the maximum error on a logarithmic scale. Figure 6 Intuitively, as the iteration level increases from 0 to 9, the total number of spline nodes increases from the initial 100 to 1064 as needed, and the corresponding maximum error exhibits an exponential decay characteristic. This convergence curve proves that the node dynamic encryption strategy of this invention can accurately locate high-frequency error intervals and allocate parameters as needed, achieving sub-picometer-level design accuracy that surpasses the traditional fitting limit with extremely high parameter utilization.

[0098] 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 this invention disclosure can be executed in parallel, sequentially, or in different orders, as long as the desired result of the technical solution disclosed in this invention can be achieved, and this is not limited herein.

[0099] The specific embodiments described above do not constitute a limitation on the scope of protection of this invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this invention should be included within the scope of protection of this invention.

Claims

1. A method for designing CGH fixtures for complex surfaces based on spline modeling, characterized in that, include: S1: Determine the sampling points within the effective aperture range of the CGH, and determine the target phase for each sampling point; S2: Using B-spline functions as the mathematical representation of the target phase, construct the current node model; using the sampling points and their target phase obtained in step S1, optimize the current node model to obtain the optimal node model and the optimal node matrix corresponding to the optimal node model. The process of constructing the current node model in step S2 includes: S21: From all the sampling points in step S1, uniformly select a small number of sampling points as nodes, and construct the initial spline function of the nodes with phase as output and sampling point coordinates as input; S22: Calculate the difference between the initial spline function in step S21 and the target phase corresponding to all sampling points to obtain the initial global residual distribution mapped to each node interval; S23: Based on the residual distribution within each node interval obtained in step S22, find the interval to be optimized, and dynamically insert new nodes into all intervals to be optimized; S24: Merge the new node inserted in step S23 with the original node, maintaining the node density unchanged; construct the current spline function using the obtained node according to step S21. S25: Based on the current spline function obtained in step S24, construct an overdetermined system of equations with all target phases as objectives and spline coefficients as unknowns; S26: Solve the minimum norm solution of the overdetermined system of equations in step S25, obtain the spline coefficients with the minimum global residual sum of squares under the current node distribution, and obtain the current node model. The corresponding current node distribution is the current node matrix. S3: Based on the optimal node matrix obtained in step S2, output the data used for CGH encoding manufacturing.

2. The method for designing complex surface CGH fixtures based on spline modeling according to claim 1, characterized in that, In step S1: The target phase for each sampling point is determined by the following formula: ； in, i Let λ represent the target phase at the i-th sampling point, and λ represent the wavelength of the reference light emitted to the CGH. OPD i This represents the optical path difference between the reference light and the target light at the i-th sampling point.

3. The method for designing complex surface CGH fixtures based on spline modeling according to claim 1, characterized in that, Step S23, which involves dynamically inserting new nodes into all intervals to be optimized, includes: S231: Sort all intervals to be optimized in descending order of their maximum residual values ​​to obtain a sorted index sequence; S232: From the sorted intervals in step S231, select the first η% intervals to be optimized as the set to be encrypted in this round, and insert a new node at the geometric midpoint of the η% intervals to be optimized to obtain multiple new node intervals; S233: Calculate the interval width of each new node interval in step S232, and repeat the operation of steps S231 to S232 for all new node intervals whose interval width is greater than or equal to the preset minimum node spacing threshold.

4. The method for designing complex surface CGH fixtures based on spline modeling according to claim 1, characterized in that, The overdetermined system of equations in step S25 is as follows: ； Where J(c) represents the overdetermined system of equations for solving the spline coefficients c of the spline function, μ k This represents the surface shape of the k-th sampling point. target (μ k ) represents the target phase corresponding to the k-th sampling point, M represents the total number of sampling points, and c i Let N represent the i-th spline coefficient in the spline function, and let n represent the total number of spline coefficients. i,p Let represent the p-th term B-spline basis function in the spline function.

5. The method for designing complex surface CGH fixtures based on spline modeling according to claim 1, characterized in that, The stopping condition for optimization in step S2 is that the maximum residual of the currently obtained node matrix is ​​less than the preset residual threshold.

6. The method for designing complex surface CGH fixtures based on spline modeling according to claim 1, characterized in that, Step S2 also includes: using the analytical properties of B-splines and numerical iterative algorithms to verify the obtained optimal node matrix.

7. The method for designing complex surface CGH fixtures based on spline modeling according to claim 6, characterized in that, The verification process includes: Within the effective aperture range of the CGH, multiple sampling points mutually exclusive with those in step S1 are determined to construct a verification point set; Based on the obtained optimal node model, the phase gradient at any verification point within the effective aperture of the CGH is directly and analytically calculated using the derivative property of B-splines. Based on the calculated phase gradient, the reverse vector of the light rays emitted from the CGH is determined. Based on the reverse vector, the surface ray parameter equation from the CGH plane to the optical element under test is established, and the exit point of the light rays emitted from the CGH is obtained. ； in, This represents the inverse vector, and t represents the optical path parameter. This represents the spatial coordinates of the ray corresponding to the optical path parameter t. This represents the initial spatial coordinates of the ray on the CGH plane, i.e., the coordinates of the ray's exit point on the CGH plane; CGH emits light rays from the exit point onto the surface of the optical element under test, and determines the normal vector at the intersection of the surface of the optical element under test and the light rays; Construct the positive ray path from the surface of the optical element under test back to the CGH plane based on the normal vector, and obtain the return point of the return ray and the CGH plane; Calculate the geometric positional deviation and ray angle deviation between the launch point and the return point.

8. The method for designing complex surface CGH fixtures based on spline modeling according to claim 7, characterized in that, The optical path parameter is obtained by iteratively solving the following formula: ； Among them, t j Let represent the optical path parameter obtained in the j-th iteration, F represent the surface shape equation of the optical element under test, and α represent the under-relaxation factor.

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