CN116841024A - Free-form surface optical system design method with addition of turning manufacturing constraint - Google Patents
Free-form surface optical system design method with addition of turning manufacturing constraint Download PDFInfo
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- G02B17/00—Systems with reflecting surfaces, with or without refracting elements
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Abstract
The invention provides a free-form surface optical system design method added with turning manufacturing constraint, which is used for solving the technical problems that the processing difficulty of a free-form surface is ignored in the traditional free-form surface design method, so that the processing difficulty of the free-form surface is high and the surface type processing precision cannot meet the requirement. The design method of the optical system provided by the invention comprises the following steps: according to the analytic expression of the free-form surface, calculating to obtain a reference aspheric busbar equation of the free-form surface; uniformly sampling data points in the rho direction and the theta direction on the free curved surface to obtain the deviation RMS value of the sampled data points relative to the reference aspheric surface; adding the deviation RMS value of the sampled data points from the reference aspheric surface as a manufacturing constraint to an evaluation function of the optical design; the method can be used for completing the design of the free-form surface optical system by setting the target value of the corresponding evaluation function to zero and selecting the weight of the target value, and effectively reduces the processing difficulty of the free-form surface on the premise of meeting the imaging requirement.
Description
Technical Field
The invention relates to a free-form surface processing method, in particular to a free-form surface optical system design method with the addition of turning manufacturing constraints.
Background
Free-form surfaces are complex optical elements that lack translational or rotational symmetry in their surface topography, and will introduce new design freedom into the optical design compared to conventional spherical and aspherical mirrors. The design of a reflective optical imaging system using free-form surfaces instead of conventional spherical and aspherical mirrors may result in a number of advantages to be achieved by the optical system, including simplification of the system structure, reduction of the number of optical elements, improvement of imaging quality, and expansion of the imaging field of view. However, since the free-form surface does not have rotational symmetry, the processing accuracy of the free-form surface is limited in actual processing.
In the conventional freeform surface design method, indexes of an optical system, such as imaging quality, structural parameters of the optical system and the like, need to be constrained according to design requirements, and at present, only the imaging quality of the optical system is usually considered, but the processing difficulty of the freeform surface in the system is ignored. Therefore, the free curved surface is likely to be difficult to process after the optical system is designed, and even the processing accuracy of the surface is not required due to the too large processing difficulty of the surface.
Disclosure of Invention
The invention aims to solve the technical problems that the processing difficulty of a free-form surface is ignored in the traditional free-form surface design method, so that the processing difficulty of the free-form surface is high, and the surface type processing precision of the free-form surface cannot meet the requirement, and provides a free-form surface optical system design method added with turning manufacturing constraint.
In order to achieve the above object, the technical idea of the present invention is as follows:
and analyzing the machinability and machining difficulty of different surface types of the free curved surface based on a single-point turning method. When the turning process is controlled based on the fast tool servo system, the machined workpiece needs to be placed on the C-axis for high-speed rotation. In turning a free-form surface, the C-axis rotational angular velocity is usually set to a constant velocity rotation, and furthermore, the feed velocity in the x-direction is usually set to a constant value or is changed gently, so that the z-direction reciprocation of the tool bit is decisive in the machining. When the height change of the surface shape along the radial direction is smaller, the movement amplitude of the tool bit in the Z direction in the machining process is smaller, so that two benefits are brought: when the same number of points are selected on the spiral line of one period, the tangential linear interpolation error is smaller; when the reciprocating motion amplitude in the z direction is smaller, the motion precision of the linear axis is higher. Therefore, the lower the degree of deviation of the free-form surface from the reference aspherical surface, the lower the processing difficulty of the free-form surface.
Based on the above conception, the technical solution provided by the invention is as follows:
the free-form surface optical system design method for adding the turning manufacturing constraint is characterized by comprising the following steps of:
according to the analytic expression f (rho, theta) of the free-form surface, calculating to obtain a reference aspheric busbar equation of the free-form surface, wherein the specific expression is as follows:
wherein ρ represents the polar diameter in the cylindrical coordinate system, θ represents the polar angle in the cylindrical coordinate system, and h (ρ) represents the height of the reference aspheric surface corresponding to ρ in the polar diameter in the cylindrical coordinate system;
uniformly sampling data points in the rho direction and the theta direction on the free curved surface, wherein the rho direction is sampled M times and the theta direction is sampled N times; combining the reference aspheric busbar equation of the free-form surface obtained in the step 1), obtaining deviation RMS value epsilon of M x N sampling data points relative to the reference aspheric surface, wherein the expression is as follows:
where j represents the number of sampling times in the ρ direction, j=1, 2..m; i represents the number of θ -direction sampling times, i=1, 2..n; ρ j Representing the polar diameter corresponding to the rho direction sampling data point; θ i Representing a polar angle corresponding to a theta-direction sampling data point, wherein M is an integer greater than or equal to 1, and N is an integer greater than or equal to 12;
adding the deviation RMS value epsilon of M.N sampling data points relative to the reference aspheric surface as a manufacturing constraint to an evaluation function of the optical design;
and 4, setting the target value of the evaluation function corresponding to the manufacturing constraint to be zero, and selecting the weight of the evaluation function corresponding to the manufacturing constraint to complete the design of the free-form surface optical system.
Further, the steps 2 to 3 are specifically:
2, maximum caliber r of free curved surface corresponding to rho direction on free curved surface max Uniformly sampling data points in the theta direction for N times; combining the reference aspheric busbar equation of the free-form surface obtained in the step 1 to obtain an average value E' of deviation values of N sampling data points relative to the reference aspheric surface, wherein the expression is as follows:
wherein z is i Representing the height of the sampled data points;
the average value e' of the deviation values of the N sampled data points from the reference aspheric surface is added as a manufacturing constraint to the evaluation function of the optical design.
Further, in step 3 ], the evaluation function is an evaluation function in ZEMAX software.
Compared with the prior art, the invention has the following beneficial effects:
1. according to the free-form surface optical system design method added with the turning manufacturing constraint, the reference aspheric surface and the deviation RMS value relative to the reference aspheric surface are determined according to the free-form surface, so that the free-form surface turning difficulty is quantitatively described, on the basis, the deviation RMS value of the free-form surface is used as the manufacturing constraint to be added into the free-form surface design process, and compared with the traditional free-form surface design method, the free-form surface design method provided by the invention has the advantage that the free-form surface machining difficulty is effectively reduced on the premise that imaging requirements are met.
2. The free-form surface optical system design method added with the turning manufacturing constraint is simple to operate and high in practicality, and can be widely applied to mirror surface machining of free-form surfaces.
3. The invention provides a free-form surface optical system design method with addition of turning manufacturing constraint, which is characterized in that the maximum caliber r of a free-form surface corresponding to the rho direction on the free-form surface max And the data points in the theta direction are uniformly sampled, so that the form of an operand in ZEMAX software can be simplified, and the optimization speed is increased.
Drawings
FIG. 1 is a schematic diagram of a tool moving according to a control point to generate a linear interpolation error in free-form turning;
FIG. 2 is a graph showing the simulation results of the deviation RMS values of 1000 random free-form surfaces and the processed linear interpolation error RMS values.
FIG. 3 is a schematic diagram of a free-form off-axis three-reflector system without added manufacturing constraints;
FIG. 4 is a schematic diagram of a free-form off-axis three-mirror system with added manufacturing constraints;
FIG. 5 is a spot diagram and MTF plot for each field of view of a free-form off-axis three-mirror system without added manufacturing constraints, where (a) is the spot diagram and (b) is the MTF plot;
FIG. 6 is a spot diagram and MTF plot for each field of view of a free-form off-axis three-mirror system with added manufacturing constraints, where (a) is the spot diagram and (b) is the MTF plot;
FIG. 7 is a plot of the deflection values of the primary mirror in the free-form off-axis three-mirror system without added manufacturing constraints, wherein (a) is the plot of the deflection values of the secondary mirror in the free-form off-axis three-mirror system without added manufacturing constraints, (b) is the plot of the deflection values of the three mirrors in the free-form off-axis three-mirror system without added manufacturing constraints;
fig. 8 is a graph of deflection values for a free-form surface off-axis three-mirror system with a manufacturing constraint added thereto, wherein (a) is a graph of deflection values for a primary mirror in a free-form surface off-axis three-mirror system with a manufacturing constraint added thereto, (b) is a graph of deflection values for a secondary mirror in a free-form surface off-axis three-mirror system with a manufacturing constraint added thereto, and (c) is a graph of deflection values for a three mirror in a free-form surface off-axis three-mirror system with a manufacturing constraint added thereto.
FIG. 9 is a bar graph of the off-axis three-mirror system of the free-form surface without added manufacturing constraints and the off-axis three-mirror system of the free-form surface with added manufacturing constraints with primary, secondary and offset RMS values of the three mirrors;
FIG. 10 is a bar graph comparing the RMS values of the processing linear interpolation errors of the primary mirror, secondary mirror, and triple mirror in a free-form off-axis triple system without added manufacturing constraints and a free-form off-axis triple system with added manufacturing constraints.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations.
A free-form surface optical system design method for adding turning manufacturing constraint specifically comprises the following steps:
based on the characteristics of free-form surface turning, calculating and obtaining a reference aspheric busbar equation of the free-form surface according to an analytic expression f (rho, theta) of the free-form surface, wherein the specific expression is as follows:
where ρ represents the polar diameter in the cylindrical coordinate system, θ represents the polar angle in the cylindrical coordinate system, and h (ρ) represents the height of the reference aspherical surface corresponding to the polar diameter ρ in the cylindrical coordinate system.
In the invention, the magnitude of the deviation RMS value epsilon of the free-form surface is used for evaluating the processing difficulty of the free-form surface, so that data points in the rho direction and the theta direction on the free-form surface are uniformly sampled, wherein the rho direction is sampled M times and the theta direction is sampled N times; combining the reference aspheric busbar equation of a certain free-form surface obtained in the step 1), obtaining deviation RMS value epsilon of M x N sampling data points relative to the reference aspheric surface, wherein the expression is as follows:
where j represents the number of sampling times in the μ direction, j=1, 2..m; i represents the number of θ -direction sampling times, i=1, 2..n; ρ j Representing the polar diameter corresponding to the rho direction sampling data point; θ i The polar angle corresponding to the theta-direction sampling data point is represented, M is an integer greater than or equal to 1, and N is an integer greater than or equal to 12.
To verify the deviation RMS value e of the free-form surface, the difficulty of processing the free-form surface can be measured. In this embodiment, a quadric surface is taken as a base surface, a free-form surface with an additional term of 7 th order XY polynomial is taken as an example for illustration, the normalized radius of the free-form surface in this embodiment is selected to be 1mm, and the surface shape expression is:
wherein R is 0 Represents the radius of curvature of the base surface of the free-form surface, r 0 Represents the caliber of the free-form surface, k represents the conic coefficient, i represents the degree of x in the XY polynomial, j represents the degree of y in the XY polynomial, m represents the maximum degree of x in the XY polynomial, n represents the maximum degree of y in the XY polynomial, x i y j Representing a polynomial, A i，j Represents x i y j Coefficients of the polynomial.
Randomly selecting 1000 groups of XY polynomial coefficients to construct 1000 free-form surface expressions, wherein the curvature radius R of the base surface of the free-form surface 0 The cone coefficients k and XY polynomial coefficients are determined in ranges such that the constructed free-form surface is characterized reasonably, the specific ranges being shown in table 1.
TABLE 1 base radius of curvature R of free-form surface 0 Range of conic coefficients k and XY polynomial coefficients
The deviation RMS values e and the processed linear interpolation error RMS values based on the 1000 free-form surfaces are calculated separately. Wherein the deviation RMS value E is calculated according to formula (2), and the source of the linear interpolation error is that the actual track of the tool is controlled according to the number of control points, and the deviation from the ideal curved surface exists, as shown in figure 1, the control points of the tool are s i Sum s i+1 The ideal track of the cutter is s i Sum s i+1 Curve in between, while the actual path of the tool is s i Sum s i+1 A straight line in between, because there may be machining errors in the two tracks, known as linear interpolation errors. The parameters of the machining simulation are that the machining caliber is 20mm, the number of the selected control points is 200, the pitch of the spiral lines is 10um, so that the linear interpolation error distribution and the linear interpolation error RMS value of the free curved surface can be calculated according to the coordinates of the control points.
As shown in fig. 2, the relation between the deviation RMS values e of 1000 random free-form surfaces and the linear interpolation error of the processing is that the deviation RMS values e and the linear interpolation error are positively correlated according to simulation experiment results. I.e. the larger the deviation from the RMS value e, the larger the linear interpolation error that describes the machining, i.e. the greater the machining difficulty.
In summary, it is fully feasible to evaluate the difficulty of processing the free-form surface by using the deviation RMS value e of the free-form surface, and in the design, the smaller the deviation RMS value e of the free-form surface is, the lower the difficulty of processing the free-form surface is. Therefore, in order to make the processing difficulty of the free-form surface in the optical system lower, the deviation RMS value e of the free-form surface needs to be smaller.
When the deviation RMS value is added to the evaluation function as a manufacturing constraint, a large number of data points need to be sampled and calculated according to its definition. The constraint implementation in the evaluation function is performed through operands, and too many sampling points can cause too many operands, so that the speed optimization of the optical system is slow, and the implementation of software is not facilitated. Therefore, in this embodiment, the number of operands is reduced by reducing the sampling points, so as to accelerate the speed of optimizing the optical system, specifically: maximum caliber r of free curved surface corresponding to rho direction on free curved surface max The data points in the theta direction are uniformly sampled N times, and usually N is 20 to 30, and the maximum caliber r of the free-form surface corresponding to the rho direction is selected max The size of (a) determines the specific number.
Due to the angle θ of the sampled data points i =i/n.2pi, the height z of the sampled data point according to the expression of the free-form surface i The method comprises the following steps:
z i ＝f(r max ，θ i ) (4)
the average e' of the deviation values of the N sampled data points from the reference aspheric surface is:
in the subsequent optical design process, the average value epsilon' of the deviation values of the N sampling data points relative to the reference aspheric surface is added to the evaluation function of the optical design as a manufacturing constraint. In this embodiment, the evaluation function in the ZEMAX software is used to implement the addition of the manufacturing constraint, and in other embodiments of the present invention, other software may also be used to implement the addition of the manufacturing constraint. The ZEMAX software limits the elevation of the sampled data points by SSAG operands, so when adding the manufacturing constraint, the elevation of the freeform sampled data points needs to be sampled, then the manufacturing constraint is calculated according to the average value of the elevation differences between adjacent sampled data points, and then the obtained manufacturing constraint is added into the evaluation function of the ZEMAX software.
And 4, setting the target value of the evaluation function corresponding to the manufacturing constraint to be zero, and selecting the weight of the evaluation function corresponding to the manufacturing constraint to finish the design of the free-form surface optical system.
The weight selection criteria of the evaluation function corresponding to the manufacturing constraint are related to a specific optical system, and if the weight is too large, the imaging constraint in the evaluation function is smaller, so that the imaging quality of the final optical system does not meet the design requirement. However, too small a weight may result in insignificant improvement of the processability of the free-form surface, so that several attempts are required to obtain a proper weight during actual design.
The effect of the design method of the free-form surface optical system with the addition of the turning manufacturing constraint provided by the invention is further described below by taking the free-form surface off-axis three-reflection optical system as an example.
A free-form surface off-axis three-reflection optical system without adding manufacturing constraint is used as an initial structure, and the optical system is redesigned by adding manufacturing constraint on the basis of the initial structure. The effect of manufacturing constraints on free-form surface optical system design is demonstrated by comparing the imaging effects of the optical system before and after adding the manufacturing constraints with the simulated machining errors.
Fig. 3 is a schematic structural diagram of an initial structure of a free-form surface off-axis three-reflecting system without manufacturing constraints, wherein system parameters of an imaging system are respectively as follows: the entrance pupil diameter is 40mm, the f-number is 4, and the full field of view (FOV) is 4×4. Based on the initial structure, the maximum calibers of the primary mirror, the secondary mirror and the three mirrors are respectively 20mm, 15mm and 25mm. Sampling is carried out at the maximum caliber of the three mirrors, an evaluation function is constructed according to the sagittal height difference of adjacent sampled data points, the weight is designed to be 0.01, and then a new optical system, namely a free-form surface off-axis three-mirror system added with manufacturing constraints is reproducibly optimized, as shown in fig. 4. It can be seen that the various parameters of the free-form off-axis three-mirror system, such as entrance pupil diameter, F-number, and field size, to which the manufacturing constraints are added, are unchanged, while the system structure changes, such as the pitch of the mirrors, the deflection angle of the mirrors, and the surface topography of the free-form surface.
As shown in fig. 5, the light spot diagram and the MTF curve diagram of each field of view of the free-form surface off-axis three-reflection system without adding manufacturing constraint are shown, wherein (a) in fig. 5 is the light spot diagram, and (b) in fig. 5 is the MTF curve diagram. As shown in fig. 6, in order to provide a spot diagram and an MTF graph for each field of view of the free-form surface off-axis three-reflection system to which the manufacturing constraint is added, fig. 6 (a) is a spot diagram and fig. 6 (b) is an MTF graph. By comparing the dot patterns of the two systems, it can be seen that the system imaged a slightly increased spot radius RMS, but still within airy spot, after adding manufacturing constraints. The MTF curve of each field of view of the two optical systems is kept to be about 0.7 at 90lp/mm, and the MTF curve is not different from the MTF curve of the diffraction limit. Thus, comparing the imaging quality of the two optical systems, it can be seen that the imaging quality of the optical systems is slightly reduced after adding manufacturing constraints, but the diffraction limit can still be reached.
In addition, a reference aspherical equation of the free-form surface can be calculated according to formula (1), so that the distribution of the high deviation values of the primary mirror, the secondary mirror and the three mirrors relative to the reference aspherical surface in the two optical systems can be obtained. As shown in fig. 7, fig. 7 (a) shows a deviation value distribution diagram of the primary mirror, fig. 7 (b) shows a deviation value distribution diagram of the secondary mirror, and fig. 7 (c) shows a deviation value distribution diagram of the three mirrors, in order to provide a deviation value distribution diagram of the free-form surface off-axis three-mirror system without adding a manufacturing constraint. As shown in fig. 8, in order to obtain the deviation value distribution diagram of the free-form surface off-axis three-mirror system to which the manufacturing constraint is added, fig. 8 (a) shows the deviation value distribution diagram of the primary mirror in the free-form surface off-axis three-mirror system to which the manufacturing constraint is added, fig. 8 (b) shows the deviation value distribution diagram of the secondary mirror in the free-form surface off-axis three-mirror system to which the manufacturing constraint is added, and fig. 8 (c) shows the deviation value distribution diagram of the three mirrors in the free-form surface off-axis three-mirror system to which the manufacturing constraint is added. It can be seen that in the free-form off-axis three-mirror system with added manufacturing constraints, the deviation values of all mirrors from the reference aspheric surface are small, which means that all free-form surfaces are closer to the reference aspheric surface.
The deviation RMS values e of the primary mirror, the secondary mirror, and the triple mirror in the two optical systems are calculated according to the formula (2), respectively, and the result is shown in fig. 9. In addition, at the maximum aperture of the free-form surface, 200 points are selected in a single circle to simulate the linear interpolation error distribution during turning, and then the RMS value of the linear interpolation error distribution is calculated, as shown in fig. 10. Comparing the deviation RMS value e and the machining linear interpolation error RMS value of the two optical systems, it can be seen that the machining error and the machining difficulty of the free-form surface optical system are reduced after the manufacturing constraint is added in the optical design compared with the optical system without the manufacturing constraint.
Therefore, the manufacturing constraint of the free-form surface is added to the design process of the free-form surface, and the imaging quality of the system and the processing difficulty of the surface type can be considered when the optical system is designed by using optical design software. The optical system designed by the mode can effectively reduce the processing difficulty of each free-form surface reflecting mirror under the condition of ensuring the imaging quality of the optical system.
The above is only a preferred embodiment of the present invention, and is not intended to limit the present invention, but various modifications and variations can be made to the present invention by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (3)
1. The free-form surface optical system design method for adding the turning manufacturing constraint is characterized by comprising the following steps of:
according to the analytic expression f (rho, theta) of the free-form surface, calculating to obtain a reference aspheric busbar equation of the free-form surface, wherein the specific expression is as follows:
wherein ρ represents the polar diameter in the cylindrical coordinate system, θ represents the polar angle in the cylindrical coordinate system, and h (ρ) represents the height of the reference aspheric surface corresponding to ρ in the polar diameter in the cylindrical coordinate system;
uniformly sampling data points in the rho direction and the theta direction on the free curved surface, wherein the directions are sampled M times and the theta direction is sampled N times; combining the reference aspheric busbar equation of the free-form surface obtained in the step 1), obtaining deviation RMS value epsilon of M x N sampling data points relative to the reference aspheric surface, wherein the expression is as follows:
where j represents the number of ρ -direction sampling times, j=1, 2 … M; i represents the number of θ -direction sampling times, i=1, 2 … N; ρ j Representing the polar diameter corresponding to the rho direction sampling data point; θ i Representing a polar angle corresponding to a theta-direction sampling data point, wherein M is an integer greater than or equal to 1, and N is an integer greater than or equal to 12;
adding the deviation RMS value epsilon of M.N sampling data points relative to the reference aspheric surface as a manufacturing constraint to an evaluation function of the optical design;
and 4, setting the target value of the evaluation function corresponding to the manufacturing constraint to be zero, and selecting the weight of the evaluation function corresponding to the manufacturing constraint to complete the design of the free-form surface optical system.
2. The method for designing a free-form surface optical system with addition of turning manufacturing constraints according to claim 1, wherein the steps 2 to 3 are specifically:
2, maximum caliber r of free curved surface corresponding to rho direction on free curved surface max Uniformly sampling data points in the theta direction for N times; combining the reference aspheric busbar equation of the free-form surface obtained in the step 1 to obtain an average value E' of deviation values of N sampling data points relative to the reference aspheric surface, wherein the expression is as follows:
wherein z is i Representing the height of the sampled data points;
the average value e' of the deviation values of the N sampled data points from the reference aspheric surface is added as a manufacturing constraint to the evaluation function of the optical design.
3. A free-form surface optical system design method adding turning manufacturing constraints according to claim 1 or 2, characterized in that:
in the step 3), the evaluation function is an evaluation function in ZEMAX software.
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