A free-form surface optical system design method with turning manufacturing constraints added

Abstract

The application provides a free-form surface optical system design method adding turning manufacturing constraints, which is used for solving the technical problem that the machining difficulty of the free-form surface is high and the machining precision of the surface shape cannot meet the requirements in the conventional free-form surface design method because the machining difficulty of the free-form surface is ignored in the conventional free-form surface design method. The optical system design method provided by the application is as follows: according to an analytical expression of the free-form surface, a reference aspheric base curve equation of the free-form surface is calculated and obtained; data points in the rho direction and the theta direction on the free-form surface are uniformly sampled to obtain the deviation RMS value of the sampling data points relative to the reference aspheric surface; the deviation RMS value of the sampling data points relative to the reference aspheric surface is added to an evaluation function of optical design as a manufacturing constraint; the target value of the corresponding evaluation function is set to zero, and the weight thereof is selected, so that the design of the free-form surface optical system is completed, and the machining difficulty of the free-form surface is effectively reduced under the premise of meeting the imaging requirements.

Description

Technical Field

[0001] This invention relates to a method for machining freeform surfaces, and more particularly to a method for designing a freeform surface optical system with added turning machining constraints. Background Technology

[0002] Freeform surfaces are complex optical elements whose surface morphology lacks translational or rotational symmetry. Compared to traditional spherical and aspherical mirrors, freeform surfaces introduce new degrees of design freedom into optical design. Using freeform surfaces instead of traditional spherical and aspherical mirrors in the design of reflective optical imaging systems offers numerous advantages, including simplified system structure, reduced number of optical elements, improved image quality, and expanded imaging field of view. However, because freeform surfaces lack rotational symmetry, their fabrication accuracy is limited in practical applications.

[0003] Traditional freeform surface design methods require constraining the optical system's specifications based on design requirements, such as image quality and structural parameters. Currently, only the image quality is typically considered, neglecting the fabrication difficulty of the freeform surfaces within the system. This can easily lead to high fabrication difficulty for the freeform surfaces after the optical system design is completed, and may even result in insufficient surface fabrication accuracy due to the excessive difficulty in machining the surface shape. Summary of the Invention

[0004] The purpose of this invention is to solve the technical problem that traditional freeform surface design methods neglect the processing difficulty of freeform surfaces, resulting in high processing difficulty and consequently, the surface processing accuracy of freeform surfaces does not meet the requirements. The invention proposes a freeform surface optical system design method that adds turning processing constraints.

[0005] To achieve the above objectives, the technical concept of the present invention is as follows:

[0006] Based on the single-point turning method, this paper analyzes the machinability and machining difficulty of different freeform surfaces. When controlling the turning process using a fast tool servo system, the workpiece needs to be placed on the C-axis for high-speed rotation. In turning freeform surfaces, the C-axis rotational angular velocity is usually set to a uniform speed, and the feed rate in the x-direction is generally set to a constant value or changes gradually. Therefore, the reciprocating motion of the tool head in the z-direction plays a decisive role in the machining process. When the radial height variation of the surface is relatively small, the amplitude of the tool head's movement in the z-direction during machining will be smaller. This brings two advantages: when selecting the same number of points on a spiral line of one cycle, the linear interpolation error in the tangential direction is smaller; and when the reciprocating motion amplitude in the z-direction is relatively small, the motion accuracy of the linear axis is higher. Therefore, the lower the deviation of the freeform surface from the reference aspherical surface, the lower the machining difficulty of the freeform surface.

[0007] Based on the above concept, the technical solution provided by this invention is as follows:

[0008] A method for designing a freeform surface optical system with added turning machining constraints, characterized by the following steps:

[0009] 1. Based on the analytical expression of a freeform surface The equation of the reference aspherical generatrix of the freeform surface is calculated, and its specific expression is as follows:

[0010]

[0011] in, Represents the polar radius in cylindrical coordinates. Represents the polar angle in cylindrical coordinates. This indicates that the polar radius in cylindrical coordinates is The corresponding reference aspherical height;

[0012] 2) For freeform surfaces direction and Data points in the direction are sampled uniformly, where Sampling M times in the direction, The direction is sampled N times; combined with the generatrix equation of the reference aspherical surface of the freeform surface obtained in step 1), the deviation RMS values ​​of M*N sampled data points relative to the reference aspherical surface are obtained. Its expression is:

[0013]

[0014] in, express The sequence number of the direction sampling number, express The sequence number of the direction sampling number, express The polar radius corresponding to the directional sampling data point; express The polar angle corresponding to the direction sampling data point, where M is an integer greater than or equal to 1 and N is an integer greater than or equal to 12;

[0015] 3】The RMS values ​​of the deviations of M*N sampled data points relative to the reference aspherical surface It is added as a manufacturing constraint to the evaluation function of optical design;

[0016] 4. Set the objective value of the evaluation function corresponding to the manufacturing constraints to zero, and select the weights of the evaluation function corresponding to the manufacturing constraints to complete the design of the freeform surface optical system.

[0017] Furthermore, steps 2-3 are as follows:

[0018] 2) On a free surface Maximum diameter of freeform surface corresponding to direction Place, to Uniform sampling of data points in the direction Combining the generatrix equation of the reference aspherical surface of the freeform surface obtained in step 1, the average deviation of N sampled data points relative to the reference aspherical surface is obtained. Its expression is:

[0019]

[0020] in Indicates the height of the sampled data point;

[0021] 3) Average the deviations of N sampled data points relative to the reference aspherical surface. It is added as a manufacturing constraint to the evaluation function of optical design.

[0022] Furthermore, in step 3, the evaluation function is the evaluation function in the ZEMAX software.

[0023] The advantages of this invention compared to the prior art are as follows:

[0024] 1. This invention provides a design method for a freeform surface optical system with added turning manufacturing constraints. Based on the freeform surface, a reference aspherical surface and its deviation RMS value relative to the reference aspherical surface are determined to quantitatively describe the difficulty of turning the freeform surface. On this basis, the deviation RMS value of the freeform surface is added as a manufacturing constraint to the design process of the freeform surface. Compared with the traditional freeform surface design method, this invention effectively reduces the manufacturing difficulty of the freeform surface while meeting the imaging requirements.

[0025] 2. The present invention provides a design method for a freeform surface optical system with added turning machining constraints. It is simple to operate, highly practical, and can be widely used in the mirror processing of freeform surfaces.

[0026] 3. The present invention provides a design method for a freeform surface optical system with added turning machining constraints, on a freeform surface... Maximum diameter of freeform surface corresponding to direction Place, to Uniform sampling of data points in different directions simplifies the form of operands in ZEMAX software, thereby accelerating optimization. Attached Figure Description

[0027] Figure 1A schematic diagram illustrating the linear interpolation error generated by the tool moving according to the control points during free-form surface turning.

[0028] Figure 2 This is a schematic diagram of the simulation results of the deviation RMS values ​​of 1000 random freeform surfaces and the RMS values ​​of the linear interpolation error during processing.

[0029] Figure 3 A schematic diagram of an off-axis three-reflection system for a free-form surface without manufacturing constraints;

[0030] Figure 4 A schematic diagram of a free-form surface off-axis three-reflection system with added manufacturing constraints;

[0031] Figure 5 The image shows the spot pattern and MTF curve of the free-form off-axis three-mirror system without manufacturing constraints, where (a) is the spot pattern and (b) is the MTF curve.

[0032] Figure 6 The image shows the spot pattern and MTF curve of the free-form off-axis three-mirror system with added manufacturing constraints, where (a) is the spot pattern and (b) is the MTF curve.

[0033] Figure 7 The diagrams show the deviation distribution of a free-form surface off-axis three-mirror system without manufacturing constraints. (a) shows the deviation distribution of the primary mirror in the free-form surface off-axis three-mirror system without manufacturing constraints, (b) shows the deviation distribution of the secondary mirror in the free-form surface off-axis three-mirror system without manufacturing constraints, and (c) shows the deviation distribution of the three mirrors in the free-form surface off-axis three-mirror system without manufacturing constraints.

[0034] Figure 8 The diagrams show the deviation distribution of a free-form surface off-axis three-mirror system with added manufacturing constraints. (a) shows the deviation distribution of the primary mirror in the free-form surface off-axis three-mirror system with added manufacturing constraints, (b) shows the deviation distribution of the secondary mirror in the free-form surface off-axis three-mirror system with added manufacturing constraints, and (c) shows the deviation distribution of the three mirrors in the free-form surface off-axis three-mirror system with added manufacturing constraints.

[0035] Figure 9 A bar chart showing the RMS deviations of the primary mirror, secondary mirror, and third mirror in a free-form surface off-axis three-mirror system without manufacturing constraints and in a free-form surface off-axis three-mirror system with manufacturing constraints.

[0036] Figure 10 A bar chart comparing the RMS values ​​of the linear interpolation error of the primary mirror, secondary mirror, and third mirror in a free-form surface off-axis three-mirror system without manufacturing constraints and in a free-form surface off-axis three-mirror system with manufacturing constraints. Detailed Implementation

[0037] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0038] A design method for a freeform surface optical system with added turning machining constraints includes the following steps:

[0039] 1. Based on the characteristics of freeform surface turning, and according to the analytical expression of freeform surfaces... The equation of the reference aspherical generatrix of the freeform surface is calculated, and its specific expression is as follows:

[0040]

[0041] in, Represents the polar radius in cylindrical coordinates. Represents the polar angle in cylindrical coordinates. This indicates that the polar radius in cylindrical coordinates is The corresponding height of the reference aspherical surface.

[0042] 2】The deviation RMS value of the freeform surface used in this invention The size of the surface is used to evaluate the processing difficulty of a freeform surface. Therefore, the processing difficulty of a freeform surface is... direction and Data points in the direction are sampled uniformly, where Sampling M times in the direction, The direction is sampled N times; combined with the generatrix equation of the reference aspherical surface of a certain freeform surface obtained in step 1), the RMS values ​​of the deviations of M*N sampled data points relative to the reference aspherical surface are obtained. Its expression is:

[0043]

[0044] in, express The sequence number of the direction sampling number, express The sequence number of the direction sampling number, express The polar radius corresponding to the directional sampling data point; express The polar angle corresponding to the direction sampling data point, where M is an integer greater than or equal to 1 and N is an integer greater than or equal to 12.

[0045] To verify the deviation of the freeform surface from the RMS value This can be used to measure the processing difficulty of a freeform surface. This embodiment uses a quadratic surface as the base surface and a freeform surface with an additional term of a 7th-order XY polynomial as an example. In this embodiment, the normalized radius of the freeform surface is selected as 1 mm, and its surface shape expression is:

[0046]

[0047] in, The radius of curvature of the base surface of the freeform surface is represented by . Let represent the diameter of the freeform surface, k represent the conic coefficient, i represent the degree of x in the XY polynomial, j represent the degree of y in the XY polynomial, m represent the maximum degree of x in the XY polynomial, and n represent the maximum degree of y in the XY polynomial. express The coefficients of a polynomial.

[0048] 1000 sets of XY polynomial coefficients were randomly selected to construct 1000 freeform surface expressions, where the base surface radius of curvature of the freeform surface is... The ranges of the conic coefficient k and the XY polynomial coefficients are determined in order to make the constructed freeform surface reasonable in terms of characterization. The specific ranges are shown in Table 1.

[0049] Table 1. Radius of curvature of the base surface of a freeform surface The range of conic coefficients k and the coefficients of the XY polynomial.

[0050]

[0051] Calculate the deviation RMS value based on each of these 1000 freeform surfaces. And the RMS value of linear interpolation error during processing. Among them, the deviation from the RMS value... The linear interpolation error is calculated according to formula (2), and the source of the error is that the actual tool trajectory is controlled according to the number of control points, which deviates from the ideal surface. Figure 1 As shown, the control points of the tool are and The ideal trajectory of the cutting tool is and The curve between, while the actual tool trajectory is and The straight line between the two trajectories, due to the machining error between them, is called linear interpolation error. The machining simulation parameters are a machining diameter of 20mm, 200 selected control points, and a spiral spacing of 10um. Therefore, based on the coordinates of the control points, the linear interpolation error distribution and the RMS value of the linear interpolation error of the freeform surface can be calculated.

[0052] like Figure 2 The figure shows the deviation RMS values ​​of 1000 random freeform surfaces. The relationship between linear interpolation error and machining error, based on simulation results, shows a positive correlation. That is, when the error deviates from the RMS value... The larger the value, the greater the linear interpolation error in the machining process, which means the greater the machining difficulty.

[0053] In summary, the deviation from RMS value using freeform surfaces Using this method to evaluate the fabrication difficulty of freeform surfaces is entirely feasible. In the design process, the deviation of the freeform surface from the RMS value... The smaller the value, the easier it is to fabricate the freeform surface. Therefore, to make the fabrication of freeform surfaces in optical systems easier, it is only necessary to reduce the deviation of the freeform surface from its RMS value. Smaller is fine.

[0054] When adding deviations from the RMS value as manufacturing constraints to the evaluation function, it requires sampling and calculating a large number of data points according to its definition. Since constraints are implemented through operands in the evaluation function, an excessive number of sampling points leads to an excessive number of operands, which slows down the optimization of the optical system and is detrimental to software implementation. Therefore, this embodiment reduces the number of operands by reducing the number of sampling points, thereby accelerating the optimization of the optical system. Specifically, on the freeform surface... Maximum diameter of freeform surface corresponding to direction Place, to Uniform sampling of data points in the direction Typically, N is selected from 20 to 30, depending on... Maximum diameter of freeform surface corresponding to direction .

[0055] Due to the angle of the sampling data points According to the expression for freeform surfaces, the height of the sampled data points for:

[0056]

[0057] Then the average deviation of the N sampled data points relative to the reference aspherical surface for:

[0058] .

[0059] 3. In the subsequent optical design process, the average value of the deviation of N sampling data points relative to the reference aspherical surface is calculated. Manufacturing constraints are added to the evaluation function of the optical design. In this embodiment, the evaluation function in ZEMAX software is used to add manufacturing constraints. In other embodiments of the invention, other software can also be used. ZEMAX software uses the SSAG operand to limit the sag of the sampled data points. Therefore, when adding manufacturing constraints, the sag of the freeform surface sampled data points needs to be sampled. Then, the manufacturing constraint is calculated based on the average sag difference between adjacent sampled data points, and the obtained manufacturing constraint is added to the evaluation function in ZEMAX software.

[0060] 4. By setting the objective value of the evaluation function corresponding to the manufacturing constraints to zero and selecting the weights of the evaluation function corresponding to the manufacturing constraints, the design of the freeform surface optical system can be completed.

[0061] The selection criteria for the weights of the evaluation function corresponding to manufacturing constraints are related to the specific optical system. If the weights are too large, the imaging constraints in the evaluation function will be smaller, resulting in the final imaging quality of the optical system failing to meet the design requirements. On the other hand, if the weights are too small, the improvement in the processing performance of the freeform surface will not be significant. Therefore, in actual design, several trials are needed to obtain a suitable weight value.

[0062] The following example, using a freeform off-axis three-mirror optical system, further illustrates the effect of the design method for a freeform optical system with added turning manufacturing constraints provided by this invention.

[0063] Using a freeform off-axis three-mirror optical system without manufacturing constraints as the initial structure, the optical system was redesigned by adding manufacturing constraints to this initial structure. The impact of manufacturing constraints on the design of freeform optical systems is demonstrated by comparing the imaging effects of the two optical systems before and after adding manufacturing constraints, as well as by simulating manufacturing errors.

[0064] in, Figure 3 This is a schematic diagram of the initial structure of a freeform off-axis three-mirror system without manufacturing constraints. The system parameters are: entrance pupil diameter of 40mm, F-number of 4, and field of view (FOV) of 4×4. Based on the initial structure, the maximum apertures of the primary mirror, secondary mirror, and third mirror are 20mm, 15mm, and 25mm, respectively. Sampling is performed at the maximum aperture of the three mirrors, and an evaluation function is constructed based on the sag difference between adjacent sampling data points, with a weight of 0.01. Then, a new optical system is derived through optimization, i.e., a freeform off-axis three-mirror system with manufacturing constraints, as shown below. Figure 4As shown, it can be seen that the parameters of the freeform off-axis three-mirror system with added manufacturing constraints remain unchanged, such as the entrance pupil diameter, F-number, and field of view size, while the system structure changes, such as the mirror spacing, mirror deflection angle, and freeform surface morphology.

[0065] like Figure 5 The image shows the spot pattern and MTF curve of a free-form off-axis three-mirror system without manufacturing constraints, for each field of view. Figure 5 (a) in the diagram is a series of light spots. Figure 5 (b) in the figure is the MTF curve. For example... Figure 6 As shown, the image displays the spot size distribution and MTF curves for each field of view of a free-form off-axis three-mirror system with manufacturing constraints. (The image shows...) Figure 6 (a) in the diagram is a series of light spots. Figure 6 (b) in the figure shows the MTF curve. By comparing the dot plots of the two systems, it can be seen that after adding manufacturing constraints, the RMS spot radius of the system's image increases slightly, but it is still within the Airy disk. The MTF curves for each field of view of both optical systems remain around 0.7 at 90 lp / mm, which is not significantly different from the MTF curve of the diffraction limit. Therefore, comparing the imaging quality of the two optical systems shows that after adding manufacturing constraints, the imaging quality of the optical system decreases slightly, but it can still reach the diffraction limit.

[0066] Furthermore, the equation of the reference aspherical surface of the freeform surface can be calculated according to formula (1), thereby obtaining the distribution of the sagittal deviations of the primary mirror, secondary mirror, and tertiary mirror in the two optical systems relative to the reference aspherical surface. Figure 7 The figure shows the deviation distribution of an off-axis three-inverse system on a freeform surface without manufacturing constraints. Figure 7 (a) in the diagram represents the deviation distribution of the primary mirror. Figure 7 (b) in the diagram represents the deviation distribution of the secondary mirror. Figure 7 In the diagram, (c) represents the deviation distribution of the three mirrors. For example... Figure 8 The figure shows the deviation distribution of the off-axis three-inverse system of a freeform surface with added manufacturing constraints. Figure 8 (a) in the figure represents the deviation distribution of the primary mirror in a free-form off-axis three-mirror system with added manufacturing constraints. Figure 8 (b) in the figure represents the deviation distribution of the secondary mirror in a free-form surface off-axis three-mirror system with added manufacturing constraints. Figure 8 In diagram (c), the deviation distribution of the three mirrors in the freeform surface off-axis three-mirror system with added manufacturing constraints is shown. It can be seen that in the freeform surface off-axis three-mirror system with added manufacturing constraints, the deviation values ​​of all mirrors relative to the reference aspherical surface are small, which means that all freeform surfaces are closer to the reference aspherical surface.

[0067] Calculate the RMS deviation values ​​of the primary mirror, secondary mirror, and tertiary mirror in the two optical systems according to formula (2). The result is as follows Figure 9 As shown. Furthermore, at the maximum aperture of the freeform surface, 200 data points were selected in a single cycle to simulate the linear interpolation error distribution during turning, and then the RMS value of the linear interpolation error distribution was calculated, as shown below. Figure 10 As shown. Comparing the deviation RMS values ​​of the two optical systems. As well as the RMS value of linear interpolation error, it can be seen that, compared with optical systems without manufacturing constraints, the addition of manufacturing constraints to the optical design reduces both the manufacturing error and the manufacturing difficulty of freeform surface optical systems.

[0068] Therefore, this invention adds manufacturing constraints to the design process of freeform surfaces, allowing for consideration of not only the imaging quality of the optical system but also the fabrication difficulty of the surface shape when designing the optical system using optical design software. The optical system designed in this way can effectively reduce the fabrication difficulty of each freeform surface mirror while ensuring the imaging quality of the optical system.

[0069] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for designing a freeform surface optical system with added turning machining constraints, characterized in that, Includes the following steps:

1. Based on the analytical expression of a freeform surface The equation of the reference aspherical generatrix of the freeform surface is calculated, and its specific expression is as follows: ； in, Represents the polar radius in cylindrical coordinates. Represents the polar angle in cylindrical coordinates. This indicates that the polar radius in cylindrical coordinates is The corresponding reference aspherical height; 2) On freeform surfaces direction and Data points in the direction are sampled uniformly, where Sampling M times in the direction, The direction is sampled N times; combined with the generatrix equation of the reference aspherical surface of the freeform surface obtained in step 1), the deviation RMS values ​​of M*N sampled data points relative to the reference aspherical surface are obtained. Its expression is: ； in, express The sequence number of the direction sampling number, express The sequence number of the direction sampling number, express The polar radius corresponding to the directional sampling data point; express The polar angle corresponding to the direction sampling data point, where M is an integer greater than or equal to 1 and N is an integer greater than or equal to 12; 3】The RMS values ​​of the deviations of M*N sampled data points relative to the reference aspherical surface It is added as a manufacturing constraint to the evaluation function of optical design; 4. Set the objective value of the evaluation function corresponding to the manufacturing constraints to zero, and select the weights of the evaluation function corresponding to the manufacturing constraints to complete the design of the freeform surface optical system.

2. The design method for a freeform surface optical system with added turning machining constraints according to claim 1, characterized in that: In step 3, the evaluation function used is the evaluation function in the ZEMAX software.

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