End-to-end automated design method for freeform surface systems

Abstract

This invention relates to an end-to-end automated design method for freeform surface systems, specifically comprising: S1, providing an initial planar system, solving for a first-order geometric structure based on the initial planar system, the first-order geometric structure comprising multiple surfaces, and determining the optical power of each surface based on the field curvature equation and the focal length equation; S2, determining the optical power of each surface and adjusting the surface position using the flat field condition and the linear astigmatism elimination equation; S3, determining a rapid automated design strategy for the first-order geometric structure, the design strategy satisfying at least one of the following three conditions: the focal length of the system is equal to a given value, the sum of the optical powers ψ of each surface is 0, and the astigmatism parameter γ is 0; and S4, constructing the freeform surface system using a surface normal correction method, further improving image quality using an iterative process of image plane correction and surface correction, and obtaining the final design result.

Description

Technical Field

[0001] This invention relates to the field of optical design, and more particularly to an end-to-end automated design method for freeform surface systems. Background Technology

[0002] In traditional optical design, designers need to participate in solving the initial system structure and try various optimization strategies to improve image quality step by step. Current automated design methods for freeform surface systems still take several minutes or even hours to complete a single automated design. While this can yield a high-quality optical system, if the result does not meet design requirements in terms of size, fabrication, and assembly, recalculation is necessary, requiring another few minutes to hours of waiting. This prolonged waiting time can easily disrupt the designer's thought process and workflow. If the speed of the automated design process could be increased several times, a high-quality optical system could be obtained directly within the time required for a single optimization iteration in traditional optical design. Summary of the Invention

[0003] In conclusion, it is indeed necessary to provide an end-to-end automated design method for freeform surface systems to overcome the aforementioned technical problems.

[0004] An end-to-end automated design method for freeform surface systems includes the following steps:

[0005] S1 provides an initial planar system, and solves a first-order geometric structure based on the initial planar system. The first-order geometric structure includes multiple surfaces, and determines the optical power of each surface based on the field curvature equation and the focal length equation.

[0006] S2, using the flat field condition and the linear astigmatism elimination equation to determine the optical power of each surface and adjust the position of the surface;

[0007] S3, determine a rapid automated design strategy for the first-order geometry, which satisfies at least one of the following three conditions: the system's focal length is equal to a given value, the sum of the optical powers ψ of all surfaces is 0, and the astigmatism parameter γ is 0; and

[0008] S4. The freeform surface system is constructed using the surface normal correction method. The image quality is further improved by using the iterative process of image plane correction and surface correction to obtain the final design result.

[0009] Compared to existing technologies, the end-to-end automated design method for freeform surface systems provided by this invention represents a novel approach to optical design, transforming traditional optical design methods. This end-to-end design process requires only simple input and no manual intervention, directly yielding an optical system with excellent imaging quality within the timeframe of a traditional single optimization. If the single end-to-end design result does not meet design requirements or still has room for performance improvement, the input parameters can be continuously modified and the end-to-end design process repeated until a system meeting the requirements is obtained. The short timeframe of a single end-to-end design process ensures that the waiting period does not interrupt the designer's thought process. Furthermore, the designer can select from a large pool of high-image-quality optical systems to obtain the final design result based on actual needs. Attached Figure Description

[0010] Figure 1 This is a schematic diagram of the initial three-reflection system structure provided in an embodiment of the present invention.

[0011] Figure 2 This is a schematic diagram of the astigmatism correction transform vector provided in an embodiment of the present invention.

[0012] Figure 3 The flowchart illustrates the end-to-end rapid automated design method provided in this embodiment of the invention.

[0013] Explanation of main component symbols

[0014] First-order geometry of the three-reflection system 100

[0015] Main lens 102

[0016] Secondary mirror 104

[0017] Three mirrors 106

[0018] Image 108

[0019] The following detailed description, in conjunction with the accompanying drawings, will further illustrate the present invention. Detailed Implementation

[0020] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0021] This invention provides an end-to-end automated design method for freeform surface systems, which specifically includes the following steps:

[0022] S1 provides an initial planar system, and solves a first-order geometric structure based on the initial planar system. The first-order geometric structure includes multiple surfaces, and determines the optical power of each surface based on the field curvature equation and the focal length equation.

[0023] S2, using the flat field condition and the linear astigmatism elimination equation to determine the optical power of each surface and adjust the position of the surface;

[0024] S3, determine a rapid automated design strategy for the first-order geometry, which satisfies at least one of the following three conditions: the system's focal length is equal to a given value, the sum of the optical powers ψ of all surfaces is 0, and the astigmatism parameter γ is 0; and

[0025] S4. The freeform surface system is constructed using the surface normal correction method. The image quality is further improved by using the iterative process of image plane correction and surface correction to obtain the final design result.

[0026] The following will describe in detail each step of the end-to-end automated design method for freeform surface systems provided by this invention.

[0027] S1 provides an initial planar system, and solves a first-order geometric structure based on the initial planar system. The first-order combined structure includes multiple surfaces, and the optical power of each surface is determined according to the field curvature equation and the focal length equation.

[0028] In this step, the position of the surfaces is described using the surface spacing *d* and the incident angle *θ*. If the principal ray of the central field of view is set as the central ray, the surface spacing *d* is the distance between corresponding points of the central ray on adjacent surfaces, and the incident angle *θ* is the angle of incidence of the central ray on each surface. Please refer to [link to relevant documentation]. Figure 1 In this embodiment, the initial planar system is a three-mirror system, which includes a primary mirror 102, a secondary mirror 104, a third mirror 106, and an image plane 108. Wherein, d1 is the distance between the primary mirror 102 and the secondary mirror 104, d2 is the distance between the secondary mirror 104 and the third mirror 106, and d3 is the distance between the third mirror 106 and the image plane 108. θ1 is the angle of incidence of the central ray on the primary mirror, θ2 is the angle of incidence of the central ray on the secondary mirror, and θ3 is the angle of incidence of the central ray on the third mirror.

[0029] Before determining the first-order geometry, the optical power of each surface is determined using the field curvature and astigmatism correction equations. In the off-axis case, using the radius of curvature to describe the optical power is insufficiently accurate; therefore, the object-image distance from the central field of view is used to express the optical power φ of each surface, where... n′ and n are the refractive indices on the front and back of a certain curved surface, respectively. i and l o These represent the image distance and object distance of the surface, respectively. When calculating the first-order geometry, the optical transfer matrix is ​​used to analyze the system; the transfer matrix of the k-th surface is T. k The transfer matrix of the k-th surface spacing is D. k ,but Where φ k Let d be the optical power of the k-th surface.k and n k Let be the distance between the k-th surface and the (k+1)-th surface, and be the refractive index, respectively. Then, if the imaging system has N surfaces, the optical transfer matrix of the system is... To satisfy a given effective focal length (EFL), the spacing d between the surfaces and the optical power φ must meet the following requirements.

[0030] The next step is to convert the surface power into specific surface parameters to obtain the first-order geometry. When using a spherical surface, which is rotationally symmetric, to establish the first-order geometry of an off-axis system, the same radius of curvature will correspond to different powers in the meridional and sagittal planes. For high-performance systems with large fields of view and apertures, the off-axis angle is often larger to eliminate obstruction, resulting in a greater difference in power between the same radius of curvature in the meridional and sagittal planes. Therefore, this embodiment uses a bangle surface in the first-order geometry. In an off-axis system, the bangle surface can have different radii of curvature in the meridional and sagittal planes. Using a bangle surface ensures that the determined power of each surface remains consistent in both planes. According to Coddington's formula, the radius of the bangle surface in the meridional and sagittal planes can be determined using the object-image distance of each surface, as shown in the following equation:

[0031]

[0032]

[0033] Where θ′ and θ are the incident angle and the exit angle, respectively, R T and R S These are the radii of curvature on the meridional plane and the sagittal plane, respectively.

[0034] S2 uses the flat field condition and the linear astigmatism elimination equation to determine the optical power of each surface and adjust the position of the surface.

[0035] In this step, field curvature is an important aberration. To obtain a flat image plane without astigmatism, the Petzval curvature needs to be zero, which means the sum of the optical powers ψ of all surfaces needs to be zero, as shown in the equation. The above are the conditions for a draw. Substitution The equation can then be obtained with the object-image distance of each surface as the variable.

[0036] In off-axis systems, linear astigmatism with respect to field of view asymmetry is often a major component of aberrations. When calculating the first-order geometry, we choose to use the equation given by Chang to eliminate this linear astigmatism with respect to field of view asymmetry, as shown in the following equation:

[0037]

[0038] Where, m k θ is the ratio of image distance to object distance on each surface. k γ is the angle of incidence of the central ray on each surface. When γ is 0, linear astigmatism due to field asymmetry is eliminated. If the object surface is not tilted, then the image plane is perpendicular to the central ray.

[0039] The above equation has the characteristic that it can be viewed as the dot product of two vectors. Taking N=3 as an example, the equation is as follows:

[0040] γ=A·B=((1+m1)m2m3,(1+m2)m3,1+m3)·(tanθ1,tanθ2,tanθ3)

[0041] Vector A is only related to the object-image distance of each surface, meaning it is closely related to the optical power of each surface; while vector B is only related to the angle of incidence. When vectors A and B are orthogonal, the linear astigmatism caused by asymmetry in the field of view is eliminated.

[0042] The following will describe the formula-based approach. The astigmatism correction transform is used in this method to correct the surface position. A vector diagram of the astigmatism correction transform is shown below. Figure 2 As shown. When the optical power of each surface is determined, vector A is also determined. In this case, vector B can only make γ equal to 0 if it lies on plane 100, which is perpendicular to vector A. The angles of the surfaces in the given initial planar system usually cannot ensure that vector B lies on plane 100. Therefore, vector B can be projected onto plane 100 using a projection transformation to obtain vector B′, as shown. Figure 2 As shown. After Figure 2 The projection transformation transforms vector B into vector B′, which is orthogonal to vector A. With vector A unchanged, the dot product γ of vectors B′ and A becomes zero, and vector B′ minimizes |BB′|. From a system structure perspective, this projection transformation, given a fixed optical power, eliminates asymmetric linear astigmatism on each surface with minimal adjustment of the incident angle. This projection transformation can also be called an astigmatism correction transformation.

[0043] S3, Determine a rapid automated design strategy for the first-order geometry, which satisfies at least one of the following three conditions: the focal length of the system is equal to a given value, the sum of the optical powers ψ of each surface is 0, and the parameter γ related to astigmatism is 0.

[0044] The input to this method includes system parameters such as focal length, field of view, and F-number, as well as a planar system according to the designer's initial concept. To ensure that the final system does not deviate too much from the designer's initial planar system, the spacing d between the surfaces remains constant when determining the first-order geometry, and the incident angle θ of the central rays on each surface only changes when using astigmatic correction transformation. Assume there are a total of N surfaces, with surface spacing d1, d2, ..., d... N It is determined according to a given planar system. The difference between the image distance of the k-th surface and the object distance of the (k+1)-th surface is known as d. k If the image distances of the first (N-1) surfaces are known, the object distances of the 2nd to Nth surfaces can be obtained. Meanwhile, the object distance of the 1st surface is a given object distance, usually infinity, and the image distance of the Nth surface is d. N Thus, the object-image distances for all surfaces can be obtained. According to the formula... The object-image distance of each surface can then be used to solve for the optical power of that surface. In summary, if the image distances of the first (N-1) surfaces are known, the optical power of each surface can be obtained. Therefore, the image distances of the first (N-1) surfaces will be set as variables for solving the relevant equations.

[0045] The first-order geometric structure design strategy of this method is to ensure that the system satisfies the following three conditions as much as possible: the focal length of the system is equal to the given value, as shown in equation [equation missing]. As shown, this invention is referred to as the focal length equation; the sum of the optical powers of each surface, ψ, is 0, as shown in the equation. As shown, this invention refers to it as the field curvature equation; the parameter γ related to astigmatism is 0, as shown in the equation. As shown, this invention refers to it as the astigmatic equation. This invention utilizes the above three equations to solve for optical power and correct the incident angles of each surface. In cases where multiple solutions occur during the equation-solving process, this invention calculates the corresponding first-order geometric structure for each solution and automatically selects the system with the best image quality for the subsequent construction of the freeform surface system.

[0046] The end-to-end automated design method for freeform surface systems provided by this invention selects different design strategies depending on the number of mirrors, and is currently applicable to systems with up to four mirrors. As mentioned above, a system containing N mirrors has (N-1) variables. When N is greater than 3, the number of variables will not be less than the number of equations, and the design strategy is to solve for the optical power of the first-order geometry by simultaneously solving three equations. When N is not greater than 3, the number of variables will be less than the number of equations, and the design strategy is to combine three conditions and astigmatism correction transformation to determine the first-order geometry.

[0047] The design strategies for first-order geometry structures of two-mirror (N=2), three-mirror (N=3), and four-mirror (N=4) will be introduced below.

[0048] First-order geometry design strategy for a three-mirror system:

[0049] Based on the description of the system variables above, there are two variables in the three-mirror system: the image distances of the first two mirrors. In this case, the number of variables will be less than the number of equations, and the first-order geometric design strategy for the three-mirror system is divided into two steps.

[0050] The first step is to solve for the optical power by simultaneously applying the position-curvature equation and the focal length equation. Then, an astigmatic correction transformation is used to correct the surface position while maintaining the solved optical power, ultimately allowing the system to simultaneously satisfy three conditions. Solving for the optical power by simultaneously applying the position-curvature equation and the focal length equation (at which point an analytical solution exists) allows the system to satisfy the first two conditions. At this point, the surface position and optical power do not satisfy the astigmatic equation, meaning γ is not zero. Then, using the astigmatic correction transformation to adjust the incident angle at the current optical power satisfies the astigmatic equation. This determines the parameters of the first-order geometric structure that simultaneously satisfies all three conditions.

[0051] Before the astigmatism correction transformation, if the absolute value of γ is relatively small, it means that the surface position is close to the position satisfying the astigmatism equation. Therefore, the incident angle adjustment caused by the astigmatism correction transformation will also be relatively small, and the system will not experience occlusion due to the adjustment of the incident angle. In this case, there is no need to proceed to the second step. This situation corresponds to some geometric structures with relatively small astigmatism, such as three-mirror anastigmatism (TMA) structures. When the absolute value of γ is relatively large, the astigmatism correction transformation will cause a significant shift in the surface position, and using the astigmatism correction transformation may cause occlusion in the system. This actually means that it may be difficult to find a solution that simultaneously satisfies all three conditions around the initial planar system. Therefore, a fast and automated search process for optical power allocation is needed as the second step in the three-mirror system design strategy.

[0052] The second step involves first establishing multiple first-order geometries with different power allocations. Then, astigmatic correction transformations are applied to these first-order geometries to obtain multiple systems with γ=0. Finally, the system with the best image quality within the unobstructed range is selected as the final first-order geometry. Different power allocations are obtained by simultaneously solving the focal length equation and the astigmatic equation with different γ values; an analytical solution exists at this point. γ is selected from 0 in both positive and negative directions with a certain step size. The astigmatic correction transformation essentially finds the geometries with small aberration potential under the condition of minimal adjustment of the incident angle. Solving using astigmatic equations with different γ values ​​does not only consider astigmatism; its essence is to obtain multiple different power allocations. Then, a series of first-order geometries with small aberration potential can be obtained through the astigmatic correction transformation. Searching for γ starting from 0 means that the surface position adjustment caused by astigmatic correction increases from small to large. The search process stops when surface occlusion occurs. This method uses the average RMS spot radius of each field of view to evaluate the imaging quality of a series of geometries, and the best results will be combined with the direct design method to construct a freeform surface system.

[0053] First-order geometry design strategy for two-mirror systems:

[0054] For a two-mirror system, if the previously described rules for selecting variables are followed, with only the primary mirror distance as the variable, it will be detrimental to solving the equations. This method, when dealing with two-mirror systems, also sets the distance between the secondary mirror and the image plane as a variable, thus enabling a strategy similar to that used in three-mirror systems to be implemented in two-mirror systems.

[0055] First-order geometry design strategy for a four-mirror system:

[0056] In a four-mirror system, after determining the surface spacing and incident angle based on a given initial planar system, the four-mirror system has three variables: the image distances of the first three mirrors. At this point, the number of variables equals the number of equations. By directly solving the focal length equation, field curvature equation, and astigmatism equation simultaneously, the optical power of each surface can be obtained, and the first-order geometry can be determined. Solving this nonlinear system of equations can be done using an optimization algorithm to calculate a numerical solution. This method uses a general global optimization algorithm in the 1stopt software to solve this problem. This method can obtain calculation results without depending on the selection of initial values, and is fast, stable, and universal, yielding correct results in most cases.

[0057] S4. The freeform surface system is constructed using the surface normal correction method. The image quality is further improved by using the iterative process of image plane correction and surface correction to obtain the final design result.

[0058] The first step is the construction of the freeform surface system. Since the first-order geometry has already undergone power allocation and angle correction based on astigmatism and field curvature, completing the correction of many low-order aberrations, using the direct design method to correct higher-order aberrations can quickly achieve excellent image quality. The direct design method used in this paper is the surface normal correction method, which can rapidly realize the construction of a freeform surface imaging system.

[0059] In this direct design method, each surface requires multiple fitting processes. By simultaneously considering the coordinates and normals, the original surface data points and the corrected normal of a given surface are fitted together to obtain a freeform surface. The original surface data points are obtained by ray tracing of characteristic rays from multiple fields of view with different apertures. The corrected normal is determined by the incident and outgoing rays according to the law of reflection, where the incident ray maintains its original direction and the outgoing ray points towards the ideal image point. Fitting each surface in the system sequentially completes the establishment of the freeform surface system. The process of fitting all freeform surfaces in the system needs to be repeated multiple times until the image quality no longer improves, thus completing the construction of the freeform surface imaging system.

[0060] Next comes the correction of the freeform surface system. The determination of the image plane angle in the first-order geometry only considers linear astigmatism due to field-of-view asymmetry. To further improve image quality, an iterative process of image plane correction and surface normal correction is required. In this iterative process, image plane correction considers aberrations related to the image plane, and surface correction also uses the direct design method of surface normal correction. Since the system's image quality is already quite good before image plane correction, it can often be completed within a few iterations. When the system's image quality meets the required standards, the final freeform surface system is obtained.

[0061] The end-to-end automated design method for freeform surface systems provided by this invention follows this process: After the designer provides system parameters and an initial planar system, the method first determines the surface position and optical power of each surface in the first-order geometry based on the flat field condition and the linear astigmatism elimination equation. Then, it uses a surface normal correction method to construct the freeform surface imaging system. Finally, it uses an iterative process of image plane correction and surface correction to further refine the freeform surface imaging system, thus obtaining the final design result. The flowchart of the end-to-end automated design method for freeform surface systems provided by this invention is as follows: Figure 3 As shown.

[0062] Furthermore, based on the final obtained freeform surface system, a mirror that meets the parameters is made and installed to obtain an optical lens element.

[0063] The automated design method proposed in this paper enables rapid end-to-end design of freeform surface imaging systems, potentially driving a change in optical design paradigms. This method is both immediate and efficient; designers can often achieve excellent design results through several rapid trial-and-error processes in the end-to-end design process. For designers familiar with various structures of freeform surface systems, it is likely that an off-axis imaging system with excellent image quality can be obtained in one or two attempts. For beginners in optical design or professionals in other fields of optics, the desired design result can usually be obtained after several trial-and-error processes.

[0064] The end-to-end design method proposed in this invention takes an initial planar system as input, and the position of the surface does not move or only moves within a small range during the automatic design process. This feature ensures that the optical path structure of the system can conform to the designer's general concept. By properly positioning the planar surface, the designer reduces the possibility of beam obstruction or lens interference. Even if these phenomena occur in the design result, the designer can avoid them by simply adjusting the position of the planar surface during trial and error.

[0065] To design a compact optical system, a loose initial planar system can be provided during the initial end-to-end design process. Based on the output system obtained from the initial end-to-end design, the designer can predict the surface shape and size of the output system in subsequent trial and error processes. The designer can then continuously change the position of the planar system to reduce its volume and achieve a compact optical structure.

[0066] Please refer to the table below, which provides a comparison between the system designed by the end-to-end automated design method for freeform surface systems provided by this invention and existing technologies.

[0067]

[0068] The table above lists several representative examples of freeform surface system design using existing automated design methods. These examples mainly include three-mirror systems and four-mirror systems. The table also provides specific embodiments of the end-to-end automated design method for freeform surface systems provided by this invention. As can be seen from the table, the end-to-end automated design method for freeform surface systems proposed in this invention is applicable to high-performance imaging systems and enables rapid, automated end-to-end design.

[0069] Furthermore, those skilled in the art may make other changes within the spirit of this invention, and of course, all such changes made in accordance with the spirit of this invention should be included within the scope of protection claimed by this invention.

Claims

1. An end-to-end automated design method for freeform surface systems, which specifically includes the following steps: S1 provides an initial planar system. A first-order geometric structure is solved based on this initial planar system. This first-order geometric structure includes N surfaces. The optical power Φ of each surface is determined according to the field curvature equation and the focal length equation. The principal ray of the central field of view is set as the central ray. The spacing d between the surfaces is the distance between corresponding points of the central ray on adjacent surfaces. The incident angle θ is the incident angle of the central ray on each surface. n' and n are the refractive indices before and after a certain curve, respectively, l i and l o These are the image distance and object distance of the curved surface, respectively. S2, using the flat-field condition and the linear astigmatism elimination equation, determines the optical power of each surface and adjusts the surface position. The spacing d and optical power Φ of each surface need to satisfy... Where EFL is the given effective focal length, and the transfer matrix of the c-th surface is T. c ; S3, Determine a rapid automated design strategy for the first-order geometry. This strategy must satisfy at least one of the following three conditions: the system's focal length is equal to a given value, the sum of the optical powers ψ of all surfaces is 0, and the astigmatism parameter γ is 0. k is an integer less than or equal to N, Φ k Let be the optical power of the k-th surface, and the linear astigmatism elimination equation be: ;as well as S4. The freeform surface system is constructed using the surface normal correction method. The image quality is further improved by using the iterative process of image plane correction and surface correction to obtain the final design result.

2. The end-to-end automated design method for freeform surface systems as described in claim 1, characterized in that, In step S1, when calculating the first-order geometry, the optical transfer matrix is ​​used to analyze the system, and the transfer matrix of the k-th surface is T. k The transfer matrix of the k-th surface spacing is D. k ,but , where Φ k Let d be the optical power of the k-th surface. k and n k denoted as the distance and refractive index between the k-th surface and the (k+1)-th surface, respectively.

3. The end-to-end automated design method for freeform surface systems as described in claim 1, characterized in that, In step S2, in the absence of astigmatism, the sum of the optical powers ψ of each surface is 0.

4. The end-to-end automated design method for freeform surface systems as described in claim 1, characterized in that, In step S3, the spacing d between the surfaces remains constant when determining the first-order geometry, and the incident angle θ of the central ray on each surface changes only when using the astigmatic correction transformation.

5. The end-to-end automated design method for freeform surface systems as described in claim 4, characterized in that, There are N surfaces, with surface spacing d1, d2, …, d… N It is determined according to a given planar system, and then the difference between the image distance of the k-th surface and the object distance of the (k+1)-th surface is known as d. k Given the image distances of the first (N-1) surfaces, we can obtain the object distances of the 2nd to Nth surfaces. Simultaneously, the object distance of the 1st surface is a given object distance of infinity. The image distance of the Nth surface is then d. N Thus, the object-image distance for each surface is obtained.

6. The end-to-end automated design method for freeform surface systems as described in claim 1, characterized in that, In step S3, the first-order geometry design steps of the three-reflector system include: First, the optical power is solved by combining the surface curvature equation and the focal length equation. Then, the astigmatism correction transformation is used to correct the surface position while ensuring the solved optical power, so that the system can satisfy the three conditions at the same time. Multiple first-order geometric structures with different optical power allocations are established. Then, astigmatic correction transformations are performed on these first-order geometric structures to obtain multiple systems with γ=0. The final first-order geometric structure is selected within the unobstructed range.

7. The end-to-end automated design method for freeform surface systems as described in claim 1, characterized in that, In step S4, the surface normal correction method includes: each surface undergoes multiple fitting processes, and by fitting the original surface shape data points and the corrected normal of a certain surface simultaneously, a free surface is obtained by fitting the original surface shape data points and the corrected normal; the original surface shape data points are obtained by ray tracing of characteristic rays with different apertures in multiple fields of view; the corrected normal is determined by the incident ray and the outgoing ray according to the law of reflection, wherein the incident ray keeps its direction unchanged and the outgoing ray points to the ideal image point.

Images