In-situ contour detection method for large-aperture quasi-planar optical components based on three probes

Abstract

The present invention relates to the field of optical detection, and in particular to a method for in-situ contour detection of large-aperture quasi-planar optical elements based on three probes, comprising the following steps: S1: planning the detection points and detection path of the large-aperture quasi-planar optical element; S2: using a three-probe detection device to scan the detection points distributed on the large-aperture quasi-planar optical element point by point according to the planned detection path; wherein the lasers emitted by the three laser probes of the three-probe detection device are respectively incident at an angle on the surface of the large-aperture quasi-planar optical element to form three measurement points, and the distance values ​​from the three laser probes to the measurement points are simultaneously obtained; S3: using the three distance values ​​to calculate the spatial coordinates of the three measurement points and perform plane fitting to obtain the two-dimensional slopes of the detection points; and S4: reconstructing the surface shape of the large-aperture quasi-planar optical element based on the two-dimensional slope of each detection point. The present invention can realize large-scale mirror contour detection and improve the accuracy of mirror contour detection.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical detection, and in particular relates to an in-situ contour detection method for a large-aperture quasi-planar optical element based on three probes. Background Art

[0002] With the rapid development of large-scale optical systems in fields such as astronomical observation, high-energy lasers, and space exploration, large-aperture quasi-flat optical components (diameters exceeding meters) are increasingly widely used. For example, the surface accuracy of core optical components in ground-based giant telescopes (such as the European Extremely Large Telescope (E-ELT), space-based gravitational wave detectors (such as LISA), and inertial confinement fusion devices (such as the National Ignition Facility (NIF)) directly determines the overall performance of the optical system. For example, the NIF's plane mirrors have an aperture of up to 1.5 meters, and the peak-to-valley (PV) accuracy of the machined surface must be better than (λ / 6 = 632.8nm). Any local micron-level profile deviation will cause laser beam wavefront distortion and, in severe cases, thermal damage to the plane mirror. These large-aperture quasi-flat optical components are susceptible to dynamic surface errors during manufacturing, assembly, and use due to factors such as gravitational deformation, stress release, and ambient temperature fluctuations. High-precision, large-aperture inspection methods are urgently needed to ensure their performance.

[0003] The machining process for large-aperture quasi-planar optical components is generally divided into stages such as roughing, milling, grinding, and polishing. Inspection methods for the grinding stage are primarily categorized into two types: contact and non-contact. Mechanical contact scanning methods, such as coordinate measuring machines (CMMs), utilize high-precision probe systems to scan and measure the mirror profile point by point, achieving sub-micron accuracy. Non-contact inspection methods primarily employ infrared interferometry, which records phase changes through interference fringes, enabling high-precision inspection. However, both types of inspection methods present certain challenges. For example, during contact inspection, the probe directly contacts the mirror surface, potentially damaging it, and the accuracy required is difficult to achieve. While interferometer-based non-contact inspection methods offer high accuracy, their detection range is limited to the standard mirror aperture, making them difficult to directly extend to the inspection of large-aperture quasi-planar optical components. Summary of the Invention

[0004] In view of this, the present invention aims to provide a method for in-situ contour detection of large-aperture quasi-planar optical components based on three probes, so as to solve the technical problem that the existing detection methods in the grinding stage cannot take into account both high precision and large detection range.

[0005] To achieve the above object, the technical solution created by the present invention is implemented as follows:

[0006] A method for detecting the contour of a large-aperture quasi-planar optical component in-situ based on three probes comprises the following steps:

[0007] S1: Plan the inspection points and inspection paths for large-aperture quasi-planar optical components;

[0008] S2: A three-probe detection device is used to scan the detection points distributed on the large-aperture quasi-planar optical element point by point according to the planned detection path. The lasers emitted by the three laser probes of the three-probe detection device are respectively incident at an angle on the surface of the large-aperture quasi-planar optical element, forming three measurement points, and simultaneously obtaining the distance values ​​from the three laser probes to the measurement points.

[0009] S3: Calculate the spatial coordinates of the three measurement points using the three distance values, and then perform plane fitting on the spatial coordinates of the three measurement points to obtain the two-dimensional slope of the detection point;

[0010] S4: Reconstruct the surface shape of the large-aperture quasi-planar optical element based on the two-dimensional slope of each detection point.

[0011] Furthermore, after obtaining the spatial coordinates of the three measurement points, a differential operation is performed on the spatial coordinates of the three measurement points to obtain the real spatial coordinates of the three measurement points after removing errors, and plane fitting is performed using the real spatial coordinates of the three measurement points.

[0012] Furthermore, let the spatial coordinates of the three measurement points be 、 、 , then the fitted plane equation is:

[0013] ;

[0014] ;

[0015] ;

[0016] ;

[0017] The two-dimensional slope of the detection point includes the slope of the plane in the x direction and the slope of the plane in the y direction :

[0018] ;

[0019] in, A, B, C represents the normal vector of the plane, and D represents the intercept of the plane.

[0020] Furthermore, step S4 specifically includes the following steps:

[0021] S41: Constructing a surface reconstruction model based on the relationship between the two-dimensional slope and the sagittal height of the detection points;

[0022] S42: Use the least squares method to solve the surface reconstruction model and obtain the vector height of each detection point Z ;

[0023] S43: The vector height of each detection point Z Combined with each other, the reconstructed surface shape of the large-aperture quasi-planar optical element is obtained.

[0024] Furthermore, the relationship between the two-dimensional slope and the vector height of the detection point is:

[0025] ;

[0026] ;

[0027] in, Indicates the detection point ( i , j )exist x The slope in the direction, Indicates the detection point ( i , j +1) x The slope in the direction, Indicates the detection point ( i , j )exist y The slope in the direction, Indicates the detection point ( i , j +1) y The slope in the direction, Indicates the detection point ( i , j ) of Yadaka, Indicates the detection point ( i , j +1) of Yagao, represents the distance between two detection points, express x Direction or y The number of detection points in the direction;

[0028] The constructed face reconstruction model is:

[0029] ;

[0030] in, K Indicates merging all detection points in x The slope in the direction y The slope in the direction column vector; S represents the coefficient matrix, which is sparse matrix of ; Z Indicates the vector height of all detection points Column vector.

[0031] Furthermore, the least squares method is used to solve the face reconstruction model:

[0032] ;

[0033] in, It is the coefficient matrix S The transpose of .

[0034] Furthermore, the three measuring points form an equilateral triangle, and the radius of the circumscribed circle of the equilateral triangle is used as the detection aperture composed of the three laser probes. The calculation formula for the detection aperture is:

[0035] ;

[0036] Among them, Ф represents the central axis of the three laser probes, a Indicates the distance between each laser probe and the central axis Ф, Indicates the angle between each laser probe and the horizontal plane, Indicates the distance value measured by each laser probe, r Indicates the distance between each measuring point and the center axis Ф, which is the size of the detection aperture.

[0037] Furthermore, the three-probe detection device includes a tool and a machine tool with a lifting function. The machine tool includes a guide rail, and the three laser probes are installed on the guide rail through the tool.

[0038] Furthermore, the large-aperture quasi-planar optical element is a plane mirror or a weakly curved mirror, and the mirror surface of the weakly curved mirror satisfies any one of the following conditions:

[0039] Condition 1: The radius of curvature R is greater than or equal to 10 times the mirror diameter D;

[0040] Condition 2: The sagittal height h from the center of the mirror to the edge of the mirror does not exceed 0.5% of the mirror diameter D;

[0041] Condition 3: The local maximum slope of the surface is no more than 1°.

[0042] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0043] 1. The present invention adopts point-by-point scanning combined with surface reconstruction to perform contour detection, which can break through the limitation of detection range and is suitable for contour detection of large-aperture quasi-planar optical components.

[0044] 2. The present invention adopts an inclined multi-probe and differential slope measurement method, which effectively eliminates the influence of the motion error of the long-stroke guide rail on the contour measurement, thereby improving the contour detection accuracy.

[0045] 3. The present invention reduces the detection aperture by increasing the height of the three laser probes, thereby further improving the contour detection accuracy.

[0046] 4. The present invention adopts a surface reconstruction model to reconstruct the surface shape of large-aperture quasi-planar optical elements, with fast solution speed and high calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] The accompanying drawings, which constitute part of the present invention, are intended to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are intended to explain the present invention and do not constitute an undue limitation of the present invention. In the accompanying drawings:

[0048] Figure 1 2 is a schematic structural diagram of a three-probe detection device according to an embodiment of the present invention;

[0049] Figure 2 Schematic diagram of the calculation principle of the detection caliber according to the embodiment of the present invention;

[0050] Figure 3 2. It is a schematic diagram of the planar distribution of three measurement points according to an embodiment of the present invention;

[0051] Figure 4 1. It is a flow chart of an in-situ contour detection method for a large-aperture quasi-planar optical element based on three probes according to an embodiment of the present invention;

[0052] Figure 5 It is a schematic diagram of the planning of the detection path according to the embodiment of the invention;

[0053] Figure 6 Schematic diagram of a face reconstruction model according to an embodiment of the present invention.

[0054] Description of the accompanying drawings: machine tool 1, tooling 2, laser probe 3, large-aperture quasi-planar optical element 4. DETAILED DESCRIPTION

[0055] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is 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 only used to explain the present invention and do not constitute a limitation of the present invention.

[0056] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

[0057] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention. In addition, the terms "first", "second" and the like are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, features defined as "first", "second" and the like may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

[0058] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "installed," "connected," and "connected" should be understood in a broad sense. For example, they can refer to fixed connections, detachable connections, or integral connections; they can refer to mechanical connections or electrical connections; they can refer to direct connections or indirect connections through an intermediate medium; and they can refer to internal connections between two components. Those skilled in the art can understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0059] The present invention will be described in detail below with reference to the drawings and in combination with embodiments.

[0060] The present invention provides a method for detecting the contour of a large-aperture quasi-planar optical element in-situ based on three probes, which is implemented using a three-probe detection device. The structure of the three-probe detection device is as follows: Figure 1 As shown, the three-probe inspection device includes a machine tool 1, a fixture 2, and a laser probe 3. Machine tool 1 is a machine tool with a lifting function and includes long-stroke guide rails. Three laser probes 3 are mounted on fixture 2, which is mounted on the long-stroke guide rails. The long-stroke guide rails adjust the horizontal position of the laser probes 3 to scan the large-aperture quasi-planar optical element 4 below the laser probes 3. The lifting function of machine tool 1 adjusts the vertical height of the laser probe 3 relative to the large-aperture quasi-planar optical element 4. For example, machine tool 1 is a gantry milling machine, with the milling head and milling cutter replaced by fixture 2 and laser probe 3. Laser probe 3 uses a laser displacement sensor.

[0061] The three laser probes 3 are respectively arranged tilted relative to the large-aperture quasi-planar optical element 4 so that the lasers emitted by the three laser probes 3 are incident obliquely on the surface of the large-aperture quasi-planar optical element 4. The light spot positions of the laser probes 3 on the surface of the large-aperture quasi-planar optical element 4 are used as measurement points, forming a total of three measurement points. By adjusting the tilt angles of the three laser probes 3, the geometric figure formed by the lines connecting the three measurement points is an equilateral triangle. The radius of the circumscribed circle of the equilateral triangle is the detection aperture formed by the three measurement points.

[0062] One purpose of tilting the three laser probes 3 is to adjust the height of the three laser probes 3, thereby adjusting the size of the detection aperture. The smaller the detection aperture, the higher the detection accuracy. The present invention can adjust the size of the detection aperture based on the mirror detection accuracy.

[0063] The calculation method of detection caliber is as follows Figure 2 and Figure 3 As shown, let the center axis of the three laser probes be Φ, and measure the distance from each laser probe to the center axis Φ. a The angle between each laser probe and the horizontal plane , the distance measured by each laser probe is The distance from each measuring point to the central axis Ф can be calculated from the following geometric relationship: r :

[0064] ;

[0065] Calculated r That is the size of the detection aperture.

[0066] Another purpose of tilting the three laser probes 3 is to further improve the anti-interference ability of the three-probe detection device. The guide rail error projection for:

[0067] ;

[0068] in, Indicates the guide rail error, With the tilt angle The guide rail error projection decreases with the increase of along with decreases with the decrease of , thus increasing the tilt angle , which can enhance the anti-interference ability of the three-probe detection device.

[0069] The tilt setting of the three laser probes can also improve the detection sensitivity of small height changes. :

[0070] .

[0071] The invention provides a method for detecting the contour of a large-aperture quasi-planar optical element in-situ based on three probes, such as Figure 4 As shown, the following steps are included:

[0072] S1: Plan the inspection points and inspection paths for large-aperture quasi-planar optical components.

[0073] Large-aperture quasi-planar optical elements are plane mirrors or weakly curved mirrors. Weakly curved mirrors refer to those that deviate slightly from the ideal plane and can be approximated as local planes. Mirrors that meet any of the following conditions can be considered weakly curved mirrors:

[0074] Condition 1: The radius of curvature R is greater than or equal to 10 times the mirror diameter D (i.e. R ≥ 10D);

[0075] Condition 2: The sagittal height h from the center of the mirror to the edge of the mirror does not exceed 0.5% of the mirror diameter D (i.e. h / D ≤ 0.005);

[0076] Condition 3: The local maximum slope of the surface is no more than 1°.

[0077] For ease of presentation, both flat mirrors and slightly curved mirrors are referred to as the inspection mirror. The optimal inspection point distribution scheme is determined for different inspection mirror specifications and mirror inspection accuracy requirements. Specifically, different inspection apertures are selected based on the size and material properties of the inspection mirror. The number of inspection points and path planning are determined based on the size of the inspection mirror and the inspection aperture.

[0078] In a specific example of the present invention, the number of detection points is 25, and the distance between two adjacent detection points is 4 cm.

[0079] S2: A three-probe detection device is used to scan the detection points distributed on the large-aperture quasi-planar optical element point by point according to the planned detection path; wherein, the lasers emitted by the three laser probes of the three-probe detection device are respectively incident at an angle on the surface of the large-aperture quasi-planar optical element to form three measurement points, and the distance values ​​from the three laser probes to the measurement points are obtained at the same time.

[0080] The detection path is planned in a conventional way, such as Figure 5 As shown, the inspection path is a raster scanning path that winds and bends multiple times from a first node at one end of the inspection mirror to a second node at the other end. The first and second nodes are symmetrical about the center of the inspection mirror. The scanning direction of the three-probe inspection device is indicated by the arrows, scanning the inspection mirror point by point, switching to the next row after completing a scan.

[0081] The present invention actually uses a point-by-point scanning method to complete the mirror surface contour detection of the mirror to be inspected. The advantage of gradual scanning is that it can break through the limitation of the detection range and is suitable for the contour detection of large-aperture quasi-planar optical elements.

[0082] The distance values ​​measured by the laser probe are screened for abnormal data. Abnormal data refers to data with large fluctuations caused by machine tool movement, or invalid values ​​(such as 0 or null) obtained due to the laser probe not responding in time during the inspection process. The screening method is to subtract the three data points obtained simultaneously by the three laser probes from each other. If the subtraction result is greater than the PV value of the surface shape of the mirror to be inspected or the subtraction result is invalid, the data set is discarded.

[0083] S3: Calculate the spatial coordinates of the three measurement points using the three distance values, and then perform plane fitting on the spatial coordinates of the three measurement points to obtain the two-dimensional slope of the detection point.

[0084] Assume that the distance values ​​measured by the three laser probes are , the inclination angle between the optical axis of each laser probe and the vertical direction , the azimuth angle of each laser probe in the horizontal plane , the spatial coordinates of each measurement point are calculated according to the following formula:

[0085] .

[0086] The spatial coordinates of the three measurement points can be calculated by the above formula The spatial coordinates of the three measurement points All include guide rail errors Since the guide rail error is usually a low-frequency change, it can be approximated as the same constant between two adjacent laser probes. The height difference of the three measuring points satisfies:

[0087] ;

[0088] in, ,and ; Indicates the measurement point and measuring points The height difference measured between Indicates the measurement point and measuring points The actual height difference between Indicates the measurement point The guide rail error, Indicates the measurement point Guide rail error.

[0089] like is approximately constant, then .

[0090] Due to guide rail error In the difference operation, they are canceled by subtraction. Only reflects the height difference of the true outline of the mirror , and the guide rail error Therefore, the spatial coordinates of the three measurement points are Through the above-mentioned differential operation, the influence of the guide rail error can be eliminated, thereby improving the contour detection accuracy.

[0091] The present invention fundamentally breaks through the reliance of traditional contour detection methods on high-precision guide rails by performing differential operations on the spatial coordinates of the measurement points, making it possible to achieve in-situ contour measurement accuracy of submicron order using ordinary machine tools. This breaks through the difficult trade-off between measurement efficiency, environmental adaptability and local defect sensitivity in traditional detection technologies, and realizes submicron precision in-situ dynamic detection of large-aperture plane mirrors.

[0092] The plane equation obtained by plane fitting is:

[0093] ;

[0094] ;

[0095] ;

[0096] ;

[0097] Then the two-dimensional slope of the detection point is:

[0098] ;

[0099] in, A, B, C represents the normal vector of the plane, D represents the intercept of the plane, represents the slope of the plane in the x direction, Represents the slope of the plane in the y direction.

[0100] S4: Reconstruct the surface shape of the large-aperture quasi-planar optical element based on the two-dimensional slope of each detection point.

[0101] Step S4 specifically includes the following steps:

[0102] S41: Construct a surface reconstruction model based on the relationship between the two-dimensional slope and the sagittal height of the detection point.

[0103] The detection point and the point to be reconstructed are located at the same position, and each detection point has slope measurement values ​​in two directions, that is, the slope values ​​in two directions measured by the three-probe detection device at each detection point.

[0104] The relationship between the two-dimensional slope and the vector height of each detection point is:

[0105] ;

[0106] ;

[0107] in, Indicates the detection point ( i , j )exist x The slope in the direction, Indicates the detection point ( i , j +1) x The slope in the direction, Indicates the detection point ( i , j )exist y The slope in the direction, Indicates the detection point ( i , j +1) y The slope in the direction, Indicates the detection point ( i , j ) of Yadaka, Indicates the detection point ( i , j +1) of Yagao, represents the distance between two detection points, express x Direction or y The number of detection points in the direction.

[0108] Since the data collected by the detection points are discrete, the surface reconstruction model can be expressed in the form of a matrix:

[0109] ;

[0110] in, K Indicates merging all detection points in x The slope in the direction y The slope in the direction column vector; S represents the coefficient matrix, which is sparse matrix of ; Z Indicates the vector height of all detection points Column vector.

[0111] S42: Use the least squares method to solve the surface reconstruction model and obtain the vector height of each detection point Z .

[0112] The formula for solving the face reconstruction model using the least squares method is:

[0113] ;

[0114] in, It represents the transpose of the coefficient matrix S.

[0115] S43: The vector height of each detection point Z Combined with each other, the reconstructed surface shape of the large-aperture quasi-planar optical element is obtained.

[0116] The present invention adopts a surface reconstruction model to reconstruct the surface shape of a large-aperture quasi-planar optical element without the need for iteration, and has a fast solution speed and high calculation efficiency.

[0117] It should be understood that the various forms of the processes shown above can be used to reorder, add, or delete steps. For example, the steps described in the present disclosure can be performed in parallel, sequentially, or in a different order, as long as the desired results of the technical solutions disclosed in the present disclosure can be achieved. This is not limited herein.

[0118] The above specific embodiments do not limit the scope of protection of the present invention. Those skilled in the art will appreciate that various modifications, combinations, sub-combinations, and substitutions may be made based on design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention are intended to be included within the scope of protection of the present invention.

Claims

1. A method for in-situ contour detection of large-aperture quasi-planar optical components based on three probes, characterized in that: The steps include: S1: Plan the inspection points and inspection paths for large-aperture quasi-planar optical components; S2: A three-probe detection device is used to scan the detection points distributed on the large-aperture quasi-planar optical element point by point according to the planned detection path. The lasers emitted by the three laser probes of the three-probe detection device are respectively incident at an angle on the surface of the large-aperture quasi-planar optical element, forming three measurement points, and simultaneously obtaining the distance values ​​from the three laser probes to the measurement points. S3: Calculate the spatial coordinates of the three measurement points using the three distance values, and then perform plane fitting on the spatial coordinates of the three measurement points to obtain the two-dimensional slope of the detection point; S4: reconstructing the surface shape of the large-aperture quasi-planar optical element according to the two-dimensional slope of each detection point; step S4 specifically includes the following steps: S41: Constructing a surface reconstruction model based on the relationship between the two-dimensional slope and the sagittal height of the detection points; S42: Use the least squares method to solve the surface reconstruction model and obtain the vector height of each detection point Z ; S43: The vector height of each detection point Z Combined, the reconstructed surface shape of the large-aperture quasi-planar optical element is obtained; The relationship between the two-dimensional slope and the sagittal height of the detection point is: ； ； in, Indicates the detection point ( i , j )exist x The slope in the direction, Indicates the detection point ( i , j +1) x The slope in the direction, Indicates the detection point ( i , j )exist y The slope in the direction, Indicates the detection point ( i , j +1) y The slope in the direction, Indicates the detection point ( i , j ) of Yadaka, Indicates the detection point ( i , j +1) of Yagao, represents the distance between two detection points, express x Direction or y The number of detection points in the direction; The constructed face reconstruction model is: ； in, K Indicates merging all detection points in x The slope in the direction y The slope in the direction column vector; S represents the coefficient matrix, which is sparse matrix of ; Z Indicates the vector height of all detection points column vector; The formula for solving the face reconstruction model using the least squares method is: ； in, It is the coefficient matrix S The transpose of .

2. The in-situ contour detection method for large-aperture quasi-planar optical elements based on three probes according to claim 1 is characterized in that: After obtaining the spatial coordinates of the three measurement points, a differential operation is performed on the spatial coordinates of the three measurement points to obtain the real spatial coordinates of the three measurement points after removing errors, and plane fitting is performed using the real spatial coordinates of the three measurement points.

3. The in-situ contour detection method for large-aperture quasi-planar optical elements based on three probes according to claim 1 is characterized in that The spatial coordinates of the three measurement points are 、 、 , then the fitted plane equation is: ； ； ； ； The two-dimensional slope of the detection point includes the slope of the plane in the x direction and the slope of the plane in the y direction : ； in, A, B, C represents the normal vector of the plane, and D represents the intercept of the plane.

4. The in-situ contour detection method for large-aperture quasi-planar optical elements based on three probes according to claim 1 is characterized in that: The three measuring points form an equilateral triangle. The radius of the circumscribed circle of the equilateral triangle is used as the detection aperture composed of the three laser probes. The calculation formula for the detection aperture is: ； Among them, Ф represents the central axis of the three laser probes, a Indicates the distance between each laser probe and the central axis Ф, Indicates the angle between each laser probe and the horizontal plane, Indicates the distance value measured by each laser probe, r Indicates the distance between each measuring point and the center axis Ф, which is the size of the detection aperture.

5. The in-situ contour detection method for large-aperture quasi-planar optical elements based on three probes according to claim 1 is characterized in that: The three-probe detection device includes a tooling and a machine tool with a lifting function. The machine tool includes a guide rail, and three laser probes are installed on the guide rail through the tooling.

6. The in-situ contour detection method for large-aperture quasi-planar optical components based on three probes according to claim 1 is characterized in that: The large-aperture quasi-planar optical element is a plane mirror or a weakly curved mirror, and the mirror surface of the weakly curved mirror meets any of the following conditions: Condition 1: The radius of curvature R is greater than or equal to 10 times the mirror diameter D; Condition 2: The sagittal height h from the center of the mirror to the edge of the mirror does not exceed 0.5% of the mirror diameter D; Condition 3: The local maximum slope of the surface is no more than 1°.

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