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Abstract
Magnetorheological finishing (MRF) is a deterministic optical processing technique based on CCOS that achieves high removal efficiency and processing accuracy while reducing subsurface damage. This technique still suffers from multiple iterations of processing due to variations in removal efficiency and the inability to fully correct mid-frequency errors below the cut-off frequency of the removal function. For the above problems, this paper attempted to establish the error model of removal function efficiency change for predicting the change of MRF efficiency. Based on the analysis of the distribution of surface shape residuals under different machining paths, a process combining spiral scanning and raster scanning is proposed, which can realize the correction of surface shape and restrain the deterioration of mid-frequency errors. The experimental results show that when the low-frequency errors of fused silica element surface converge rapidly, by optimizing the machining removal coefficient and using the spiral scanning and raster scanning combined method, the PSD analysis results show that the mid-frequency errors of the combined process is lower than the initial value, which expands the process route for the MRF of high-precision optical elements.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid expansion of high-precision optical system application scenarios, the precision requirements of optical elements used in high-end optical systems such as extreme ultraviolet lithography, earth satellite observation, and synchrotron radiation light source are also constantly increasing [1–3], the traditional polishing process is no longer sufficient to meet the precision requirements of high-end optical components. MRF is a sub-aperture processing technology based on CCOS, which is currently the main technical means for manufacturing precision optical components [4–5], which is widely used in the polishing of strong optical components due to its advantages of high processing accuracy, small subsurface damage, high removal efficiency, and so on. MRF is a technique that utilizes magnetorheological fluid's rheological effect in a gradient magnetic field to form a flexible polishing ribbon at the bottom of the polishing wheel, which can achieve material removal [6]. The removal function, as the realization medium of MRF, characterizes the amount of material removed per unit time when the ribbon is applied to the surface of the optical element with a certain indentation depth. Therefore, the state of the removal function and the removal efficiency play a decisive role in the polishing effect.
However, MRF still has certain shortcomings. Yang [7] found that the removal function efficiency during processing fluctuates with the depletion of the polishing particles in the polishing solution, the removal efficiency cannot be guaranteed to be stable and constant during long-time processing. Based on the principles of computer-controlled sub-aperture machining technology, MRF polishing requires that the polishing device traverse the entire surface of the workpiece along a specific path. These errors are manifested in the surface topography as regular micro machining lines, that is, mid-frequency errors. Hu and Zhou [8–10] found that the increase in the number of iterations in MRF processing results in a greater variety of main components contributing to surface errors, with a corresponding increase in the proportion of mid-frequency errors. When the frequency of the mid-frequency errors exceeds the cut-off frequency of the removal function, these errors cannot be removed efficiently due to the limitation of the magnetorheological trimming capability.
Different from the conventional evaluation criteria such as PV and RMS that only focus on the surface of optical components, for high-end optical components, more attention should be paid to the mid-frequency errors of the processed surface. Studies have shown that errors in different frequency bands have different effects on the performance of optical components. Low-frequency errors mainly affect the imaging of the optical system and introduce various aberrations; mid-frequency errors cause small-angle scattering of the incident light, which produces flares that damage the surface of the optical element; and high-frequency errors cause large-angle surface scattering of the beam and body scattering of the incident light, which affects the reflectivity of the optical system [11]. To unify the standard for measuring the high-frequency errors on the surface of optical components, ISO introduced the power spectral density (PSD) analysis index in the ISO 10110 standard issued in 1997 to evaluate the surface shape error of optical components. The corresponding characteristic evaluation curve equation is as follows:
In the formula, $f$ is the spatial frequency, $A$ is a constant, $B$ is the frequency power index, C and $D$ are the minimum and maximum spatial period. The most widely used is the evaluation curve proposed by Lawrence Livermore National Laboratory (LLNL) for NIF, which is mainly used for the analysis of mid- and high-frequency errors in the 0.03mm∼8.3 mm-1 band on the surface of an optical element [12], with the equation:
To suppress the mid- and high-frequency errors caused by MRF processing, researchers have done a lot of research on the different causes of errors in this frequency band. In the process of optical machining, it is generally believed that when the removal function is processed along the irregular removal trajectory, a “self-correction process” will be generated, that is, the more chaotic the processing trajectory, the smaller the mid-frequency errors [13]. Based on the above principles, Christina R. Dunn [14] from Zeeko proposed a path planning method based on pseudo-randomness, which added the uncertainty of the tool path during the process of the tool traversing the entire surface to be machined, so as to suppress the mid-frequency errors brought by regular path planning. Dai [13] proposed a concept based on entropy increase, trying to superimpose a random disturbance of a certain amplitude in the direction perpendicular to the scanning motion, so as to achieve the suppression of mid-frequency errors. Wang [15] proposed a theoretical analysis model of the non-stationary effect of removal function in MRF and determined the optimal material removal amount corresponding to the non-stationary effect of removal function through simulation, thus realizing the suppression of the mid-frequency errors. In the later processing stage, the texture changes brought by the polishing process can be eliminated with a very small amount of removal by smoothing technology, so as to eliminate the influence of high-frequency errors. The Optical Center of the University of Arizona adopted the stress disk smoothing technology when machining the GMT astronomical telescope, the mid-frequency errors converged from 4.9 nm to 4.6 nm [16]. Yin [17] adopted shear thickening polishing to achieve the ability to suppress the mid-frequency errors and improve the high-frequency roughness of the 300 mm planar X-ray mirror. Wang [18] developed a new “chemistry enhanced shear thickening” (C-STP) process, using Fenton reagents to achieve finished finishes with sub-surface 10 nm damage at twice the rate of conventional STP polishing.
Researchers have achieved significant breakthroughs in suppressing mid-frequency errors through various process methods and intervention times, but there are still some shortcomings. The method of pseudo-random path planning can indeed achieve the effect of suppressing mid-frequency errors, but this randomly varying path planning puts high demands on the dynamic performance of the machine tool, and ordinary equipment is unable to realize the huge acceleration and deceleration speeds required at the instant of commutation of the polishing device along the random path. Processes such as optical smoothness, on the other hand, suffer from deterioration of low-frequency errors in the process of correcting mid-frequency errors as well as low convergence efficiency. To solve the above problems, we try to use a combination of raster scanning paths and spiral scanning paths based on the “self-correction” principle of disordered processing. In the case of the dynamic performance requirements of the machine tool remain unchanged, so that the two orderly but unrelated machining paths are superimposed on each other, and the residuals of the original single path on the surface of the workpiece are intersected with each other to break the regularity of the path. Convergence suppression of errors in the mid-frequency bands is achieved while ensuring that the RMS correction of the low-frequency surface shape errors are to $\mathrm{\lambda /100\ (\lambda =\ 632}\textrm{.8nm)}$.
2. Theoretical analysis
2.1 Establishment of error model of efficiency change of removal function
According to the principle of deterministic machining, MRF process only needs to obtain the surface errors distribution of the workpiece to be machined by measuring means such as wavefront interferometer or profilometer and calculate the removal amount per unit time of the removal function. Then the residence time at any position of the workpiece surface can be calculated.
The current mainstream processing theory assumes that the removal efficiency of the removal function is constant during processing, which means the removal efficiency used to solve the residence time does not change. However, the efficiency of the removal function during actual processing decreases with the increase of the liquid processing cycle time [7]. In order to avoid inaccuracies in the removal efficiency of the removal function leading to an increase in processing iterations and subsequent deterioration of mid-frequency errors on the surface, the following mathematical model is attempted under the condition of no consideration of changes in other parameters:
Assuming that the initial surface shape evaluation index of the workpiece is $Rm{s_1}$, “pre-machining” is introduced before formal machining. In the “pre-machining”, the workpiece is processed with a small removal amount $X$, and the evaluation index of the measurement surface shape after machining is $Rm{s_2}$. Theoretically, the following formula should be satisfied between the removal amount $X$ and the front and back shape evaluation index:
At this point, there exists a corresponding relationship between the theoretical removal amount $X$ aand the actual removal amount $x$. By utilizing this relationship, assuming the evaluation index of the final machined surface of the workpiece is, it is possible to calculate the accurate removal amount $Y$ for the next processing step. Its calculation formula is as follows:
The above model guides the determination of the error relationship between the theoretical removal amount and the actual removal amount in the previous machining step, in order to direct the precise adjustment of removal parameters for the second iteration. This approach allows for the neglect of errors resulting from variations in the efficiency of the removal function, thereby achieving a highly efficient machining process with fewer iterations.
2.2 Establishment and simulation of raster – spiral path combination process model
As shown in Fig. 2, the main path planning of MRF can be divided into raster scanning path and spiral scanning path. In the actual processing process, both path planning methods need to determine a relatively appropriate discrete spacing (generally no more than 1/6 of the width of the removal function [19]) by analyzing the size of the removal function, so as to achieve high convergence efficiency and obtain high-precision and low-defect optical element surface. However, these methods also have some disadvantages that cannot be ignored. Initially, both the raster scanning path and the spiral scanning path fall under the category of sub-aperture correction techniques, thereby introducing new mid-frequency errors during the machining process. Additionally, the adoption of a single discrete step in the raster scanning may cause the emergence of sharp peaks at certain frequency bands on the PSD analysis curve of the surface of post-machined optical elements. Second, for raster scanning path, the polishing tool speed in the transverse direction can change continuously, but the line speed will have a larger mutation, is the important reason for the error caused by periodic mid-frequency ripple [20].
When spiral scanning path machining is adopted, the workpiece rotates around the axis driven by the spindle, and the polishing tool only needs to move along the bus of the workpiece, without the situation of speed mutation such as line feed, and the error correction effect of the workpiece edge is better than that of the raster scanning. However, since each point on the workpiece rotates at the same angular velocity, the centerline velocity of the workpiece is almost zero, which requires the machine tool to cross the center point at an infinite speed. When the dynamic performance of the machine tool cannot meet the requirements, it will cause over machining at the center of the workpiece. At the same time, the spiral scanning path also needs to solve the problem of tool alignment between the center of the removal function and the center of the workpiece. Because the removal function of MRF is an inverted D-shaped spot (as shown in Fig. 1), it is difficult to find the center point in the machining process. Referring to the machining process of ultra-precision lathes, when the center of the workpiece and the center of the turning tool are not in the same contour line, there may be a sharp point or a boss in the center of the workpiece [21–22], as shown in Fig. 3. Moreover, due to the limitation of workpiece rotating processing, the spiral scanning path is not suitable for the processing of non-rotating symmetric components, which limits the application scope of this processing method to a certain extent.
To solve the above problems, this paper tries to use the raster and spiral scanning path combination method. First, the spiral scanning path is used to modify the shape in a certain proportion. At this time, the surface of the workpiece will appear a ring grain, and the center may appear a certain high or low defect, as shown in Fig. 3. After that, the raster scanning path is used to correct the remaining surface residuals, and the center low/high point problem can be solved. In this regard, this paper simulates the residual distribution of the surface shape under the processing only using the raster scanning path, the processing only using the spiral scanning path, and the superposition of the raster path and the spiral path, as shown in Fig. 4.
The simulation results show that the residual errors after raster scanning path machining are obviously distributed in horizontal parallel, and the residual errors after spiral scanning path machining are obviously distributed in ring, and there is a high point in the center of the workpiece that cannot be corrected. The simulation residuals with the superposition of the two show a granular “fragmentation” distribution of errors, and the high points generated by the spiral scanning path are corrected by the subsequent raster scanning path.
Previously, we mentioned the “self-correction process” of optical processing, that is, the messier the processing path, the better the suppression effect of mid-frequency errors. The chaotic processing path can be represented by the degree of confusion of the error distribution on the surface of the optical element. In 1948, Shannon [23] proposed to use information entropy to represent the uncertainty of the existence state of things. Therefore, according to the above principle, we tried to calculate the entropy value of residual errors on the surface of optical components in the X/Y direction after simulation, respectively, to compare the chaos degree of residual errors on the surface of optical components under the three paths planning. For optical components, the distribution of surface residual error is the size of Z value in three-dimensional coordinates, so the occurrence frequency of different Z values after simulation is statistically analyzed, and the entropy value is determined by calculating the following formula:
In this formula, $\sum\limits_{i = 1}^n {{p_i} = 1} $, ${p_i}$ is the frequency of occurrence of different Z values.
According to Eq. (5), the entropies of the surface error distribution of the optical element after the simulation of the three paths are calculated respectively, the results are shown in Fig. 5.
The calculation results show that the entropy of the combined process is greater than that of the two traditional path planning methods in both directions. Therefore, it is clear that the surface confusion of the optical element after the combination process is higher, and the suppression effect of the combined process on the mid-frequency error is better than that of the conventional process.
3. Experiment
To verify the correctness of the removal model and the raster and spiral scanning combined process simulation model, a self-developed MRF machine was used for comparison test. The experimental objects were four 40 mm diameter fused silica samples to be processed, numbered G1, G2, G3 and G4. G4 is used to verify the error correction model of removal efficiency variation, and the rest is used to verify the correctness of the combined process. The initial PV and RMS index, path selection and convergence target of the surface shape of each test piece are shown in Table 1.
Table 1. Experimental item parameters and path selection
Because the experiment involves the selection of different process routes, to prevent parameters other than path planning from affecting the experimental results, the relevant process parameters of this experiment are set uniformly as shown in Table 2.
Table 2. Experimental technological parameter
4. Experimental result
4.1 Experimental result of the error model of removing function efficiency variation
To verify the correctness of the error correction model for the change of removal function efficiency, a two-week follow-up experiment was conducted on the change of removal function efficiency. The specific operation is: in two weeks, the removal function is randomly proposed for the liquid in the process, and its removal efficiency is solved. A total of 10 removal functions are extracted this time (as shown in Fig. 6). It can be observed that with the increase of processing time, the transverse width of the removal function decreases to a certain extent. MRF is a typical range of material removal. In the process of machining, the size of the removal function will affect the removal efficiency. For each extracted removal function, its corresponding removal efficiency is calculated, and the efficiency curve of the function is drawn as the cycle time changes (as shown in Fig. 7).
The above experimental results show that with the iterative increase of processing time, the removal efficiency generally declines. After 23 hours of cyclic processing, the removal efficiency of the removal function is only 64% of the original. Therefore, the variation error of the removal efficiency of the separation removal function has a positive significance for reducing the number of processing times.
In the experiment, the initial RMS of the workpiece is 0.394λ, the removal amount of the first “pre-machining” machining is 30%, and the RMS converges to 0.281λ after 15 min of processing. At this time, the actual removal amount is 28.7%, which is 30% less than the set amount. The experimental results are shown in Fig 8.
Fig. 8. Comparison of errors before and after 30% removal of “pre-machining": (a) before process, (b) after process
Set the next RMS convergence target to 0.04λ. According to the calculation of Eq. (4), the removal quantity required for the second processing is 89.7%. According to the above analysis results, experiments were carried out. The RMS of the final machining results (as shown in Fig. 9) converged to 0.043λ, and the error between the theoretical results of the model and the actual results was 7%, which proved the correctness of the error removal model.
4.2 Validation of the combined process
According to the conclusion of experiment 4.1, it is used to guide the processing of this experiment. The processing time and iterations of each test piece are shown in Table 3.
Table 3. The number and time of workpiece processing iterations
After processing, the surface shape errors distribution of each experimental piece are shown in Fig. 10.
Fig. 10. Errors distribution before and after machining:(a) before raster scanning, (b) after raster scanning, (c) before spiral scanning, (d) after spiral scanning, (e) before combined process, (f) after combined process
According to the above experimental arrangement, the following experimental results were obtained. As shown in Fig. 10(a) and Fig. 10(b), PV of the component after raster scanning path processing converges from 0.819λ to 0.053λ, RMS converges from 0.192λ to 0.008λ, and some horizontal raster lines can be seen in the middle of the workpiece. As shown in Fig. 10(c) and Fig. 10(d), PV of the component processed by the spiral scanning path converges from 1.521λ to 0.135λ, and the RMS converges from 0.394λ to 0.010λ. There is a slight fluctuation of the high point at the center of the workpiece, indicating that there is still a certain height difference between the center of the removal function and the center of the workpiece rotation, It is difficult to eliminate the high point only by using the spiral scan path and may require new adjustments to the machining tool or path planning. As shown in Fig. 10(e) and Fig. 10(f), after combined scanning path processing, component PV converges from 0.479λ to 0.035λ, RMS converges from 0.135λ to 0.006λ, and there is no obvious sharp point in the center of the workpiece after processing, and the overall surface shape shows a more obvious “fragmentation” in the center area of the workpiece, and the edge of the workpiece still has a certain ring pattern. The results are in good agreement with the simulation results.
As shown in Figs. 11(a) and 11(b) show that the low-frequency errors converge quickly, the mid-frequency errors are not obvious, and the high-frequency errors deteriorate lightly. The results of PSD analysis in both X and Y directions show that there is an obvious spike in the spatial frequency band 2mm-1, which corresponds to the discrete spacing of 0.5 mm during the machining process. Figures 11(c) and 11(d) show that the low-frequency errors converge rapidly, while the mid-frequency errors all deteriorate. It is worth noting that there is no obvious spike in the PSD results in the X and Y directions, which is caused by the staggered starting points of the two spiral scans in the two machining iterations, and the slight cross of the two spiral inward tracks.
Fig. 11. PSD analysis results before and after single machining path:(a) X direction PSD result after raster scanning processing, (b) Y direction PSD result after raster scanning processing, (c) X direction PSD result after spiral scanning processing, (d) Y direction PSD result after spiral scanning processing
To ensure the rigor of the experimental results, PSD analysis was carried out on the shape before the processing of the experimental part, the shape after the end of the spiral scanning process and the shape after the end of the combined process respectively. The results are shown in Fig 12.
Fig. 12. PSD analysis results before and after combined process path: (a) X direction PSD result after spiral scanning processing, (b) Y direction PSD result after spiral scanning processing, (c) X direction PSD result after combined processing, (d) Y direction PSD result after combined processing
Figure 12(a) and 12(b) are the PSD curve comparison between the original surface shape after machining by using spiral scanning path to correct part of the surface shape errors. The results show that when the low-frequency errors are partially corrected by the spiral scanning path, the mid-frequency errors cannot be suppressed. Figure 12(c) and 12(d) are the PSD curve comparison between the following shape and the initial surface shape when machining to the target convergence requirement using the combined process. The results show that while the low-frequency errors are corrected by the combined process, the mid-frequency errors are also suppressed well.
5. Conclusion
The experimental results show that the raster-spiral combination path can mutually correct the surface defects of optical elements caused by a single scanning path, that is, the raster scanning path can correct the middle high point that the spiral scanning path cannot remove. The two scanning paths are superimposed on each other to achieve the mutual “fragmentation” of machining patterns, and realize the machining convergence of low, middle, and high frequency band errors. The main conclusions of this paper are as follows:
- 1) By putting forward the error removal model of efficiency change of removal function, fewer machining iterations are guaranteed under the same modification target. The experimental results show that the prediction error of the model is about 7%, which verifies the correctness of the model.
- 2) The combination process of spiral scanning path and raster scanning path was proposed, and the removal efficiency errors change correction model was combined with the removal function. The low-frequency errors are corrected quickly, and the mid- and high-frequency errors of the optical element surface are lower than the initial processing value. It can meet the performance requirements of optical components in most precision optical systems.
Funding
National Natural Science Foundation of China (52293402).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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