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Abstract
Subaperture polishing is a key technique for fabricating ultra-precision optics. However, the error source complexity in the polishing process creates large fabrication errors with chaotic characteristics that are difficult to predict using physical modelling. In this study, we first proved that the chaotic error is statistically predictable and developed a statistical chaotic-error perception (SCP) model. We confirmed that the coupling between the randomness characteristics of chaotic error (expectation and variance) and the polishing results follows an approximately linear relationship. Accordingly, the convolution fabrication formula based on the Preston equation was improved, and the form error evolution in each polishing cycle for various tools was quantitatively predicted. On this basis, a self-adaptive decision model that considers the chaotic-error influence was developed using the proposed mid- and low-spatial-frequency error criteria, which realises the automatic decision of the tool and processing parameters. An ultra-precision surface with equivalent accuracy can be stably realised via proper tool influence function (TIF) selection and modification, even for low-deterministic level tools. Experimental results indicated that the average prediction error in each convergence cycle was reduced to 6.14%. Without manual participation, the root mean square(RMS) of the surface figure of a ϕ100-mm flat mirror was converged to 1.788 nm with only robotic small-tool polishing, and that of a ϕ300-mm high-gradient ellipsoid mirror was converged to 0.008 λ. Additionally, the polishing efficiency was increased by 30% compared with that of manual polishing. The proposed SCP model offers insights that will help achieve advancement in the subaperture polishing process.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Owing to the development of modern optical technology, ultra-precision optical components are being increasingly employed in high-power lasers, astronomical observations, and ultraviolet lithography systems [1–6]. For instance, the National Ignition Facility Project—the world’s largest laser device—has been under construction since 1994 and includes 7360 optical elements of various types and sizes [7]. The European Extremely Large Telescope (E-ELT), which was built in 2017, has a primary-mirror diameter of 39 m and is composed of nearly 800 off-axis hexagonal paraboloid mirrors [8]. Under the influence of strict requirements for the ultra-precision surface figure, large quantity demand, and compact schedule, the processing efficiency of the optical elements must be increased, and the deterministic level must be improved.
Subaperture polishing is an essential step in obtaining an ultra-precision surface. Computer-controlled optical surfacing (CCOS), which was proposed by Itek Inc., is a commonly used optical polishing technique for increasing surface-figure accuracy [9]. This technology is the basis for realizing deterministic surface-figure correction and is widely used in advanced polishing methods such as bonnet polishing, magnetorheological polishing, ion-beam polishing, and fluid-jet polishing [10–13]. However, owing to the existence of various polishing errors, the actual polishing results commonly deviate from the prediction results based on the convolution model; thus, the polishing process planning is difficult to self-adapt and relies on professional technicians, which results in poor polishing certainty and low efficiency.
For the polishing error in the subaperture polishing process, Walker et al. [14] mentioned ‘Fundamental complexity’, ‘Imperfect determinism’, and ‘Surface inaccessibility’ to emphasize the complexity of the polishing process and difficulty of predicting and planning the polishing process. At present, only a part of the typical errors with clear sources can be analyzed and compensated via physical modeling. For instance, the edge error is caused by the nonlinear contact-pressure distribution, which can be physically modeled via experimental fitting [15–17], finite-element analysis [18], or physical derivation [19]. Regarding the removal error caused by the surface curvature change, the contact pressures of different tools at complex curvature surfaces can be determined via contact mechanics analysis [20–22]. For the polishing error caused by the position error of the polisher, part of the kinematic error can be measured by a laser tracker [23] or force sensor [24] and can be compensated to some extent. The aforementioned studies played a role in improving the deterministic level and accuracy of subaperture polishing. However, as Walker mentioned, there is still a large proportion of errors outside the prediction and compensation ranges of the existing models. These errors exhibit ‘chaotic’ characteristics under the complex physical and chemical effects and are difficult to physically model [25]; however, they are vital to the actual polishing process. Therefore, it is crucial to realize chaotic-error perception and minimize its influence on the polishing process.
The lack of perception of chaotic error is the key reason why the self-adaptive decision of polishing parameters is impractical, in which the low prediction accuracy causes the planning decision to be misjudged. Planning methods based on the traditional CCOS convolution theory have been proposed; for example, researchers used Fourier’s criterion for tool influence function (TIF) selection [26–28], and another method is to select TIF according to the deconvolution optimization results [29]. However, owing to the impact of chaotic error in practice, it is challenging to accurately describe the actual convergence processing cycle using the traditional convolution model, which makes it difficult to put the existing self-adaptive theories into practice. Therefore, it is crucial to establish a self-adaptive decision model with the perception of chaotic errors, which is the key to ensuring the accuracy and efficiency of self-adaptive decision-making.
In summary, owing to the lack of theories and technical methods for quantitatively perceiving chaotic errors, the expected polishing results are not sufficiently deterministic; thus, polishing processing planning relies on manual intervention. Hence, it is important to achieve the perception of chaotic errors for realizing self-adaptive, deterministic, and efficient polishing and promoting the development of ultra-precision processing technology toward higher precision and intelligence levels.
The remainder of this paper is organized as follows. In Section 2, a statistical chaotic-error perception (SCP) model is established, and its significance is explained. In Section 3, an SCP self-adaptive decision model for high-precision and high-efficiency polishing is developed. Section 4 presents experimental results that validate the models, and Section 5 summarizes the paper.
2. Development and analysis of the SCP model
2.1 Development of the SCP model
Using the Preston equation, the removal amount can be expressed as the convolution of the TIF per unit time and dwell time [9]:
where z(x, y) represents the material removal amount, T(x, y) represents the dwell time, and ${\ast} $ denotes convolution.Generally, theoretical residual errors are limited by the TIF characteristics (e.g., size and shape) according to CCOS convolution theory in each polishing cycle. However, in addition to the theoretical residual error, a large proportion of chaotic errors with complex sources exist during the polishing process, which makes the theoretical material removal different from the actual material removal. The removal error varies randomly, and the detailed characteristics of the chaotic error are shown in Fig. 1, in which the difference between the theoretical and actual residual errors is the chaotic error. The theoretical residual errors represents the residual error limited by CCOS convolution theory.
Fig. 1. Diagram of chaotic-error characteristics; (a1, a2) initial surface figure; (b1, b2) theoretical residual error; (c1, c2) actual residual error; (d1, d2) chaotic error.
In essence, chaotic error is introduced by the variation in the removal rate of each dwell-time position under the influence of complex sources. Hereafter, the chaotic error can be considered to increase linearly with an increase in the material removal amount. We define the chaotic-error distribution under the unit removal amount as a random distribution δ, which can be expressed as δ·(T*TIF), where TIF is the tool influence function in the polishing cycle, and ${\ast} $ denotes convolution. The residual error of the workpiece after polishing can be expressed as:
where Eres represents the residual error of the workpiece, Einitial represents the initial surface figure of the workpiece, and H represents the removal depth during polishing. ${\|{} \|_2}$ represents the L2-norm optimization. Equation (2) can be simplified as follows:Hence, there exists a local area of appropriate size, in which the variation in the surface figure is far smaller than the variation in the chaotic error. In this local area, the surface figure can be approximated as a constant, and Eq. (5) can be simplified as follows:
According to the abovementioned theoretical derivation, the chaotic error in each polishing cycle can be statistically characterized, which enriches the existing convolution processing theory, as indicated by Eq. (8). In the SCP theory, the polishing results are not only limited by the TIF characteristic according to convolution theory, and the influence of the chaotic error is mathematically limited by the variance of the chaotic-error distribution and material removal amount.
2.2 Extraction of chaotic-error term C
The accuracy of the SCP model depends on the quality of the variance (C) acquisition, and the extraction of C is based on experimental data. Through data analysis before and after polishing, the reverse calculation can be performed as follows:
2.3 Significance of the SCP model
As described in Section 2.1, chaotic error was introduced to the model as a statistical constant to quantitatively express the polishing evolutionary process. Different processing techniques correspond to different chaotic error probability distributions, leading to different C values. By establishing a C-value database, we can accurately obtain the convergence characteristics of different processing techniques (e.g., small-tool polishing and magnetorheological finishing), as shown in Fig. 3.
The significance of the SCP model is indicated by the following mathematical relations:
- 1. Chaotic errors were perceived as statistically significant. Theoretical derivation proves that the coupling between the randomness characteristics of chaotic error (expectation, variance) and the polishing result follows an approximately linear relationship, which enriches the existing convolution convergence theory. The polishing accuracy can be quantitatively and accurately determined by the SCP model because each part of Eq. (7) can be quantified as shown in Fig. 3.
- 2. The SCP model proved the realization mechanism of high-precision polishing. There are two feasible ways to increase the processing accuracy: adjusting the volume removal rate of the TIF (the actual removal amount T*TIF) and the traditional approach of improving tool certainty (reducing the error rate C of the tool).
- 3. The proposed SCP model makes the prediction of the form error in each polishing cycle more accurate, allowing the quantitative statistical prediction of the form-error evolution for various tools in each polishing cycle. On this basis, a self-adaptive decision model including tool and processing-parameter selection was developed to realize the synchronous improvement of the polishing accuracy and efficiency.
3. Development of the SCP self-adaptive decision model for high-precision and high-efficiency polishing
According to the SCP model, a self-adaptive decision model for high-precision and high-efficiency polishing (SCP self-adaptive decision model) based on the perception of chaotic fabrication errors was developed. The optimization equation of the model is given by Eq. (10), where d represents the path interval. In the model, the proportional estimation method is used to calculate the polishing time T(d) accurately and efficiently, as follows:
Equation (10) has three variables: the removal depth H, path interval d, and TIF. The solution strategy of our model is to calculate the optimal removal depth H under different path intervals and TIF combinations separately; the parameters combination that meets the conditions of two proposed error criteria is determined to be the optimal process route.
For the solution of the removal depth H, when H increases, T(d)*TIF increases and Einitial + Hset − T*TIF remains essentially unchanged; consequently, Eopt increases. When the H decreases, T(d)*TIF remains unchanged and Einitial + Hset − T*TIF decreases owing to the limitation of T(d)min; consequently, Eopt increases. Therefore, Eq. (10) is a convex function (a single extreme point), and the optimal removal depth H can be calculated using the golden-section method. Compared with the classical gradient-descent method, the golden-section method does not require derivation and can achieve a shorter solution time [31]. The principles of the gradient-descent and golden-section methods are shown in Figs. 4(a) and (b), where $\alpha $ represents the empirically settable iteration step, derivative represents the gradient, f represents the equation to be optimized, a and b represent the set starting points, and the function value is recalculated each time the gradient-descent method is used for optimization. For a meter-scale optical component, the optimization time can be improved by a factor of >5 using the golden-section method.
To determine the optimal path interval d and optimal TIF, two criteria—the MSF and LSF errors—are proposed.
First, for the MSF error criterion, the path interval cannot be infinitely small owing to the machine speed constraint, and the removal amount for a single path point increases with the path interval. Therefore, on the premise that most of the path points on the surface figure are constrained by the velocity, the actual removal amount, T(d)*TIF, decreases with an increase in the path interval d; however, according to the convolution theorem, a more visible MSF error appears when the path interval is larger [32]. Therefore, it is important to implement self-adaptive decisions on path intervals to ensure high-precision polishing without visible MSF errors.
The MSF error caused by the path interval in polishing (MSFactual) can be expressed as:
For the comb function, the zero-frequency information of the inverse Fourier transform is the average of all the signals in the spatial domain; hence, the zero-frequency information can be deleted. In addition, the MSF error carried by the initial surface figure MSFtheory can be determined via band-pass filtering. If the conditions in Eq. (10) are satisfied, the MSF error ripple can be suppressed:
D depends on the actual tolerance of MSF error in polishing, the value of D is generally varied from 0 to 1; the smaller D indicates a more stringent MSF error tolerance.
Next, for the LSF error criterion, the optimal TIF can be selected according to the convergence efficiency R of the TIF per unit time, which is defined by the RMS values of the initial (Einitial_RMS) and final (Efinal_RMS) surface figures:
For the same surface figure, a higher value of R corresponds to a better LSF polishing ability of the TIF, indicating that the TIF is more worthy of consideration. On the premise that the process meets the MSF constraint, the TIF with the highest convergence efficiency (R) is taken as the optimal TIF.
According to the aforementioned methods, high-precision and efficient polishing can be achieved using the total iterative process, as shown in Fig. 6.
Fig. 6. SCP self-adaptive decision model flowchart for high-precision and high-efficiency polishing.
In summary, for a surface figure of the optical component to be polished, the most appropriate TIF, path interval, and optimal removal depth can be adaptively selected by the SCP self-adaptive model to achieve ultrahigh-precision and high-efficiency polishing.
4. Experiments and measurements
4.1 Experimental setup and conditions
A group of experiments was conducted using a robot polishing machine to verify the accuracy of the proposed SCP model and effectiveness of the self-adaptive decision method. An ASEA Brown Boveris (ABB) IRB 6620 robotic polisher was used, and the flange end of the robot was equipped with a polishing tool, as shown in Fig. 7(b). Several 100 mm diameter K9 mirrors with thicknesses of 10 mm were used as the experimental workpieces, as shown in Fig. 7(c). The tools consisted of foam silicone and a polishing cloth, as shown in Fig. 7(a); the other experimental conditions are presented in Table 1. Zigzag path is used in the experiments and the form error measurement is realized by laser interferometer (VeriFireXPZ, Zygo).
Table 1. Experimental conditions
4.2 Overview of experiments
Comparative experiments between the proposed SCP model-guided and traditional method-guided polishing were performed. The accuracy and efficiency of the SCP and SCP self-adaptive decision models were verified. A human–computer competition was designed to verify the improvement of the polishing efficiency with the guidance of the SCP model, in which the tool was fixed and the processing parameters were set by the traditional empirically guided process (experienced human engineers) or proposed SCP self-adaptive decision model (computer). An experiment involving multi-tool combination polishing fully guided by the SCP model was designed to verify the improvement in polishing accuracy. The comparative experiments and items to be verified are presented in Table 2.
Table 2. Comparative experiments
In addition, a validation experiment involving a practical engineering task (ϕ300-mm-aperture ellipsoid mirror) was performed to be discussed in Section 4.7. The optimal processing parameters required for these experiments were determined using the SPC self-adaptive decision model.
4.3 Error-rate extraction experiments
Two groups of experiments were performed to determine the error rates of the tools used in Experiments II and III. The initial surface figures used in Experiment I are shown in Figs. 8(a) and (c), and the polishing results are shown in Figs. 8(b) and (d). The tools were formed by two different types of foamed silicone with different elastic moduli to realize different normal forces, as shown in Fig. 7(a); the volume removal rates of the TIFs and TIF-specific formation conditions are presented in Table 3.
Fig. 8. Tool used in Experiment II: (a) initial surface figure and (b) actual polishing result. The tool used in Experiment III: (c) initial surface figure and (d) actual polishing result.
Table 3. Polishing parameters for the formation of TIFs
According to Eq. (8) and the convergence of the surface figure during the polishing process, the chaotic error rate (C) for each of the two tools was calculated, as shown in Table 4. The C values were 0.082 and 0.099 for tools A and B, respectively, and these two parameters were used in the following experiments.
Table 4. The calculation of the components of Eq. (8) for the extraction of chaotic error term C
4.4 Improvement of the polishing efficiency with the guidance of the SCP self-adaptive decision model
To verify the polishing-efficiency improvement with the guidance of the SCP self-adaptive decision model, polishing-efficiency comparison experiments were performed. Two approximately identical initial surface figures were used, as shown in Figs. 9(a) and 10(a). The polishing parameters of tool A used in the comparison experiments are presented in Table 3.
Fig. 9. Experimental results for traditional empirically guided polishing: (a) initial surface figure; (b) 1st polishing result; (c) 2nd polishing result; (d) 3rd polishing result; (e) 4th polishing result; and (f) 5th polishing result.
Fig. 10. Experimental results for SCP self-adaptive decision model-guided polishing: (a) initial surface figure; (b) 1st polishing result; (c) 2nd polishing result; and (d) 3rd polishing result.
The results of five traditional empirically guided polishing method are shown in Fig. 9. After the third polishing cycle, the surface figure no longer converged, and recurrence occurred. The removal depths for traditional empirically guided polishing are presented in Table 5.
Table 5. Removal depths for traditional empirically guided polishing
The results of the proposed SPC self-adaptive decision model-guided polishing are shown in Fig. 10. After the third polishing step, the result was better than that of the traditional method. The removal depths for polishing guided by the SCP self-adaptive decision model are presented in Table 6.
Table 6. Optimal removal depths for polishing guided by the SCP self-adaptive decision model
To illustrate the improvement in polishing efficiency with the guidance of the SCP self-adaptive decision model, a comparison map of the polishing accuracy and time for the two methods is presented in Fig. 11.
As shown in Fig. 11, for the same initial surface figure and TIF, the SCP self-adaptive decision model-guided polishing process was more efficient in achieving the same or a higher polishing accuracy; the polishing efficiency was increased by almost 30% compared with that for empirically guided polishing. This is a significant improvement, where the amount of time saved will increase as the size of the optics increases. The results confirm the effectiveness of the SCP self-adaptive decision model for polishing-efficiency improvement.
4.5 Improvement of the polishing accuracy with the guidance of SCP self-adaptive decision model
According to the SCP model, another way to achieve high-precision polishing is to adjust the volume removal rate of the TIF (actual material removal amount T*TIF). Therefore, to verify the improvement of the polishing accuracy with the guidance of the SCP self-adaptive decision model, a model-guided high-precision polishing experiment was performed for an initial surface figure, as shown in Fig. 12(a). Furthermore, a series of TIFs with different volume removal rates were taken by the robotic polisher with Tool B, as shown in Table 7.
Fig. 12. Initial surface figure; (b) 1st polishing result; (c) 2nd polishing result; (d) 3rd polishing result; and (e) 4th polishing result.(f) regional MSF error.
Table 7. TIFs to be selected for polishing and their formation parameters
Through the SCP self-adaptive decision model described in Fig. 6, the initial surface figure was polished in four cycles by three TIFs to achieve the final accuracy, as shown in Fig. 12(e). The polishing results are shown in Fig. 12, in which there were no path visible MSF ripples. To avoid the influence of interferometer noise in ultra-high precision state, the experimental results in Fig. 12 were processed via 5 × 5 median filtering and the average of 50 measurement results. The parameters of the three TIFs used in the experiment are presented in Table 7.
The optimal TIF and removal depth were determined using the SCP self-adaptive decision model for each polishing, as shown in Table 8. For only the robotic small-tool polisher, the ultimate convergence accuracy was achieved after the fourth polishing, and the RMS of the surface figure converged to 1.788 nm, which only the original MSF errors existed in initial surface figure is left. Theoretically, with a further reduction in the MSF error, the surface-figure accuracy can be further improved through appropriate TIF selection.
Table 8. Optimal TIF and removal depth for polishing guided by the SCP self-adaptive decision model
4.6 Prediction accuracy of the SCP model for each experimental polishing processing cycle
To further evaluate the prediction accuracy of the SCP model for each polishing cycle, three experimental results presented in Sections 4.4 and 4.5 were analyzed. The prediction error PE is defined by Eq. (14). The prediction results of the traditional convolution CCOS theory and proposed SCP models, actual polishing results, and prediction errors of the three experiments described in Sections 4.4 and 4.5 are presented in Table 9.
Where APR represents the RMS of the actual polishing result, and MPR represents the RMS of the model prediction result.Table 9. Predicted and actual polishing results (RMS)
As shown in Table 9, the proposed SCP model has a significantly reduced average prediction error in each convergence cycle (6.14%) compared with that of the traditional model (509%), indicating that the chaotic error level in each polishing cycle was accurately predicted by the SCP model. A more intuitive histogram is shown in Fig. 13, where the red bars (predictions of the SCP model) are consistent with the actual polishing result curves, and the blue bars (predictions of the traditional model) differ from the actual results owing to the lack of perception of the chaotic-error part. The length differences of the error bars indicate the superiority of the SCP model.
Fig. 13. Precision comparison chart of the traditional and proposed SCP models: (a) traditional empirically guided polishing experiments presented in Section 4.4; (b) proposed SCP model-guided experiments presented in Section 4.4; (c) ultrahigh-accuracy polishing experiment with the guidance of the proposed SCP model presented in Section 4.5.
Therefore, the polishing accuracy can be predicted quantitatively and precisely by the SCP self-adaptive decision model for an initial surface figure and TIF, and the average prediction error in each convergence cycle is significantly reduced compared with that of the traditional model.
4.7 Model validation for practical engineering task for each experimental polishing processing cycle
An experiment involving an aspheric ellipsoidal mirror was conducted to verify the effectiveness of the SCP self-adaptive decision model. The diameter of the workpiece was 300 mm, curvature radius of the mirror was 365.2176 mm, and K-coefficient of the mirror was –0.5463. An effective aperture of 260 mm was used. In this experiment, more polishing tools were adopted in the SCP decision model, including a 60 mm small tool, 40 mm small tool, and magnetorheological tool. The error rate of the magnetorheological tool was calibrated via the above method and set as 0.015.
The aspheric mirror is shown in Fig. 14(a), and the initial surface figure is shown in Fig. 14(b). After a cycle each of 60 mm small-tool polishing, 40 mm small-tool polishing, smoothing, and three cycles of magnetorheological polishing, the final RMS of the form error was 0.008 λ, as shown in Fig. 14(c). For each polishing cycle, the polishing tool was selected using the proposed SCP self-adaptive decision model. The removal depths for polishing guided by the SCP self-adaptive decision model are presented in Table 10.
Fig. 14. Large-aperture ellipsoid mirror: (a) material object; (b) initial surface figure; (b) final polished surface figure.
Table 10. Optimal removal depths for polishing guided by the SCP self-adaptive decision model
A polishing flowchart is shown in Fig. 15. The polishing accuracy improved in each polishing cycle with the guidance of the SCP self-adaptive decision model. Remarkably, during the fourth polishing cycle, because the MSF error became the main component of the surface error, the surface figure did not continue to converge for each polishing parameter predicted by the SCP model. Hence, a smoothing process was implemented to reduce the MSF error. The polishing parameters for the entire polishing process were determined by the SCP self-adaptive decision model. The successful convergence of the surface figure from the microscale to the nanoscale indicated that the proposed SCP model can be applied to practical engineering fabrication, which is important for improving the accuracy, efficiency, and intelligence level of optical processing.
5. Conclusion
An SCP model and a processing-parameter self-adaptive decision model were developed for quantitative prediction of the chaotic error and automatic polishing processing planning. With the guidance of the proposed model, the form-error evolution in each polishing cycle was accurately predicted for various tools, and the synchronous improvement of the polishing accuracy and efficiency was achieved via self-adaptive processing planning. The following conclusions are drawn:
- 1. Chaotic error is effectively perceived through statistical analysis rather than through physical modeling. Theoretical derivation proves that the coupling between the randomness characteristics of chaotic error (expectation and variance) and polishing results follows an approximately linear relationship, which enriches the existing convolution convergence theory.
- 2. A self-adaptive decision model considering the chaotic-error influence was developed by combining the SCP model with an LSF/MSF error criterion, which realizes the automatic decision of the optimal tool and processing parameters without manual participation. The self-adaptive decision model significantly improved the consistency and certainty of polishing.
- 3. Valuable insights into the implementation mechanism of the ultra-precision form error were obtained with the guidance of the SCP model. Compared with the existing method of improving the tool deterministic level, an ultra-precision surface with equivalent accuracy can be stably realized via proper TIF selection and modification, even for low-deterministic level tools.
Experimental results indicated that the polishing efficiency was significantly increased (by 30%) with the guidance of the SCP model (with the same or better polishing precision) and that an ultrahigh polishing accuracy was achieved without manual participation (RMS values of 1.788 nm for a flat mirror and 0.008 λ for a high-gradient ellipsoid mirror were achieved by using a small-tool robot). In addition, the average prediction error in each convergence cycle was reduced from 509% to 6.14% using the proposed SCP model compared with a traditional model.
Through the establishment of the SCP and self-adaptive decision models, our understanding of the subaperture polishing process was deepened, and a new approach for the further development of subaperture polishing is proposed.
Funding
Shanghai Sailing Program (20YF1454800); Shanghai Sailing Program (22YF1454800); National Natural Science Youth Foundation of China (62205352); Natural Science Foundation of Shanghai (21ZR1472000); Member of Youth Innovation Promotion Association of the Chinese Academy of Sciences. (2022246).
Acknowledgments
The authors would like to thank the referees for their valuable suggestions and comments that have helped improve the paper.
Disclosures
The authors declare no conflicts of interest.
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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