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Abstract
Continuous phase plates (CPPs) are increasingly being used to realize beam shaping and smoothing in high-power laser systems. With computer controlled optical surfacing (CCOS) technology, CPPs can be imprinted with high accuracy by a series of processing iterations, in which the characterization of the imprinted CPP surface plays a key role. However, the form accuracy evaluation is sensitive to the misalignment caused by the difference between the designed and measured coordinates. In this paper, the matching problem, which is the critical part of characterization, is first summarized as a least squares problem in accordance with the processing principle of CPPs. Then, the misalignment effect on the form error evaluation is analyzed. Necessary attention is paid to the CPP features and the sensibility analysis for different misalignments is conducted. To improve the efficiency and accuracy, an automatic characterization method based on image registration and nonlinear optimization is presented. Considering the smoothness of the CPP surface, the height difference tracing method is proposed to evaluate the matching performance and embedded into the characterization method. Finally, a series of simulations and experiments were undertaken to verify the performance of the proposed characterization method. The results demonstrated the feasibility of the proposed method, indicating that it can provide the reliable form error evaluation with sub-nanometer accuracy for imprinted CPPs.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is now widely recognized that continuous phase plates (CPPs) are essential diffractive optical elements that are used to realize beam shaping and smoothing in high-power laser systems [1–3]. Stringent requirements on the precision of continuously varying structured topography of CPPs are put forward for better modulation effect of the incident laser [4].
With the dwell time algorithm [5,6], computer controlled optical surfacing (CCOS) serves as a typical deterministic processing technology, which is used to improve the flatness of planar mirror or correct the aspherical or spherical optics [7]. At present, there exists many CCOS technologies for imprinting CPPs such as magnetorheological finishing (MRF) [4,7], ion beam figuring (IBF) [8], bonnet polishing (BP) [9] and atmospheric pressure plasma processing (APPP) [10,11]. However, in the field of CCOS, it is widely accepted that the high precision of optics depends on the iterative figuring process [12]. For the complexity of surface topography, the fabrication of CPPs is often a repetitive closed-loop that comprises figuring process, interferometer measurement and form error evaluation [4,8]. Given the difference in coordinates between the designed and imprinted surfaces, the form error of CPPs cannot be acquired by directly subtracting the designed surface from the measured data. Hence, freeform surface matching or registration plays an important role in the form error evaluation of imprinted CPPs, which is followed by generating the removal distribution in the next figuring progress.
Matching is generally formulated as a typical optimization problem for the best rigid transformation to minimize the objective function, and is widely used in stitching [13] and data fusion [14]. It aims to eliminate the deviations of two coordinate systems between the measurement data and the nominal template. To that end, various matching approaches have been developed to characterize the ultra-precision complex freeform surfaces. To avoid divergence, the two-phase strategy (coarse and fine) is often adopted to search for the accurate position parameters [15]. Cheung et al. [16,17] combined the five-point pre-fixture and an iterative precision adjustment algorithm with coordinate transfer to measure ultra-precision freeform surfaces. For the flat optical freeform surfaces without strong features, Kong et al. [18] introduced the coupled reference data method (CRDM) to perform high-precision characterization of the form error. Jiang et al. [19,20] proposed the structured region signature (SRS) algorithm for the initial fitting and refined the alignment based on the orthogonal distance to assess the form quality of a freeform surface. Yu et al. [21,22] presented an automatic form error evaluation method (AFEEM) for characterizing micro-structured surfaces. After the coarse registration with detected salient points, an adaptive iterative closest point (AICP) method was proposed to ensure sub-nanometer accuracy. Based on the Fourier–Mellin transform and phase correlation, Cheung et al. [23] developed an intrinsic feature-based pattern analysis method (IFPAM) for generalized form characterization. Ren et al. [24] presented a novel invariant-feature-pattern-based form characterization (IFPFC) method to address the deficiencies of conventional methods. Moreover, Liu et al. [25] investigated the any-degrees-of-freedom registration method with prior knowledge to improve the flexibility of characterization. For the large-sized micro-structured surfaces, Huang et al. [26] presented an innovative industrial-feasible method for measurement and assessment using normalized grayscale matching, which rendered the high matching accuracy and rapid processing capability.
Although strides have been made in the characterization of ultra-precision freeform surfaces, the effective and practical methods for the form error evaluation of CPPs are lacking. The specific precision requirements for the error of imprinted CPPs are in the range of tens of nanometers, which is extremely susceptible to the slight misalignment between the two surfaces. Considering the peculiarity of CPP surface, the previous characterization method cannot be directly applied. They mainly focus on the macro freeform surfaces with millimeter scale. Generally, the established optimization problem is solved with the criterion of minimum Euclidean distance. Different from this, the CPP surface is essentially quasi-planar freeform wavefronts. It can be considered as a plane superposed with multiple micron-sized waviness. For the application, the error of height normal to the plane is the key to impact the optical performance, which determines the optimization target. In this way, the flat figuring methods are adopted for the imprinting CPPs. In this procedure, the purpose of characterization not only serves as the form error evaluation, but also the generation of removal distribution for the next figuring.
This paper attempts to fill the research gap regarding the misalignment effect and the characterization method for imprinting CPPs. With the processing methods considered, the characterization problem of CPPs is first analyzed and the corresponding mathematic model is established. Then, the misalignment effect was investigated by a series of numerical simulation, which took the CPP features and the sensibility of different deviation into consideration. Next, based on the two-step strategy and the proposed height difference tracing method, the high-precision characterization method for imprinting CPPs is presented to automatically eliminate the misalignment between the designed and measured CPP surface. Finally, with the simulations and experiments, the accuracy of proposed method was evaluated to verify the capability of form error evaluation and removal generation.
2. Problem statement
2.1 Fabrication of CPPs
The method of characterization has a close connection with the corresponding processing method. Figure 1(a) shows the schematic of imprinting CPPs. Featured with varying phase profile within a scale of micrometers, the designed CPP surface is imprinted on large-aperture flat optics (430 mm × 430 mm). In this progress, the tool influence function (TIF) generated by small tools usually moves along the raster path, achieving the removal at every specific point with deterministic dwell time. However, high precision cannot be achieved with single processing.
Fig. 1. Schematic of imprinting CPPs with high precision. (a) Illustration of processing; (b) Section evolution of CPP surface.
For example, combined tools are preferred to maintain a balance between the efficiency and accuracy. With the higher removal rate, larger TIFs are usually used to imprint most topography features. Limited by the figuring capacity, only long spatial period features can be precisely generated. After that, several figuring iterations are conducted with smaller TIFs to imprint the rest of smaller spatial period features. Besides, the topographical gradient is gradually refined, which drives the performance of the CPPs in the beam line, as shown in Fig. 1(b).
Therefore, the fabrication of CPPs is a repetitive and iterative closed-loop embedded with figuring process, interferometer measurement, and form error evaluation. Before every figuring iteration, the form error evaluation is implemented and provides the removal distribution by adding the height-inverted topographical data of the designed CPP surface to the starting measured surface [8].
2.2 Principle of characterization of imprinted CPPs
In practice, the characterization cannot be conducted by directly subtracting the designed surface from the measured data. First, the measured data of imprinted CPPs are not sampled at the same interval as that of the originally designed CPP surface. The former heavily depends on the different measurement instruments. Second, the effective analyzed area of the figured surface is often smaller than the designed CPP area and needs to be accurately located at the whole designed area. Third, the measured and designed data are not both located in the same coordinate system, which results in a slight misalignment of the two datasets.
The first problem can be solved by interpolation or surface reconstruction, while the CPP matching problem in characterization comes into being with the rest of the parts, as shown in Fig. 2. For the conventional form characterization of a freeform surface, the misalignment can be eliminated by matching, which is established as an optimization problem to find the optimal rigid body transformation with six degrees of freedom (DOFs). The misalignment between the designed and measured surface is eliminated by matching to achieve the minimum Euclidean distance of corresponding point cloud pair, and can be expressed as [16],
Suppose the designed and measured surfaces are represented by discrete points, and the height along z direction is ${z_{1,i}}$ and ${z_{2,i}}$ respectively. When the misalignment is eliminated, the height difference of all the corresponding point pairs is minimized to achieve the optimal matching,
where N is the number of point pairs. Therefore, CPP matching is a typical least squares problem and can be solved by iteratively updating transformation parameters until the convergence is achieved. This part will be discussed in detail in Section 4.3. Analysis of misalignment effect for imprinting CPPs
3.1 Misalignment effect on form error evaluation
For better understanding of misalignment effect, two key definitions need firstly to be pointed. If the designed and measured CPP surfaces are given, there exist two kinds of form error. The true form error refers to the accurate height difference distribution when the two CPP surfaces are completely aligned. It is only connected with the processing error, and authentically reflects the processing quality. The analyzed form error refers to the acquired height difference distribution by matching two surfaces with specific characterization method. Ideally, the unknown true form error is expected to be predicted with the analyzed form error to conduct the form error evaluation of the imprinted CPPs. Based on this, the misalignment effect is embodied by the deviation between the true form error and the analyzed form error, which is caused by the residual misalignment. It is particularly likely to appear when the two surfaces are aligned manually.
To illustrate this, the numerical simulation was conducted with the designed CPP surface, as shown in Fig. 3. In this study, the true form error is artificially generated in this simulation. The deviation of different coordinate systems could be represented by rigid transformation matrix. Regarding to the capability of manual alignment, the residual misalignment was assumed to be Tx = 0.1 mm, Ty = 0.1 mm and rz = 0.1°in the simulation. The measured CPP surface after processing was hypothetically acquired by implementing the transformation matrix on the designed CPP surface with the added true convex form error. The analyzed form error was generated by directly subtracting the designed surface from the measured surface, which represents the manual analysis results. There was an obvious visual difference between the true and analyzed form error, which indicates the misalignment effect on form error evaluation. Besides, the root mean square (RMS) of the true and analyzed form error was 107 nm and 115 nm, respectively. Therefore, with the misalignment effect, the form error evaluation and following figuring process will be misguided.
If the designed CPP surface, true form error, and analyzed form error are denoted as S0, E0 and E, their relationship can be expressed as,
where $Misalign({\cdot} )$ is the misalignment operation and $\Delta E$ represents the form error deviation caused by the misalignment. From Eq. (7), the deviation is determined by the position misalignment of the measured CPP surface ${S_0} + {E_0}$ relative to the designed CPP surface. In practice, $RMS({{S_0}} )$ is usually several times larger than $RMS({{E_0}} )$, and dominates the degree of deviation. Hence, considering that the true form error is often unknown, the analysis of the misalignment effect focuses on the designed CPP surface ${S_0}$ shown in Fig. 3.3.2 Connection between the CPP features and misalignment effect
To characterize the typical CPP features, the Fourier spectral analysis was conducted for the designed CPP surface shown in Fig. 3, which can effectively separate the different components of CPP topography, as shown in Fig. 4. With an increase in frequency, the amplitude of single component dropped sharply. In zone A, dramatically large amplitudes were observed for every single component of different frequency, while the opposite was observed in zone B.
Two-dimensional sinusoidal waves and analysis parameters, as shown in Fig. 5(a), were defined as the topography to be simulated to investigate the influence of frequency and amplitude on the form error deviation caused by misalignment. Theoretically, there is no form error. Under the same condition of misalignment, Fig. 5(b) shows the simulated results of the RMS of form error deviation. Larger spatial frequency and amplitude were more likely to cause large deviation. On this basis, the Fourier spectral analysis in Fig. 4 was added. It was worth noting that a special contour line partially overlapped with the previous analysis result. When the frequency was larger than 0.139 mm−1 (zone B), all the components equally contributed to the form error deviation (RMS 2 nm). In zone A, larger amplitude had a stronger impact. Therefore, the form error deviation caused by misalignment is mainly attributed to the surface components with lower frequency.
Fig. 5. Connection analysis. (a) Two-dimensional sinusoidal waves; (b) Influence of spatial frequency and amplitude.
3.3 Sensibility analysis
The form error deviation caused by the misalignment effect is mainly affected by the designed CPP surface, indicating that the absolute RMS value of deviation is not suitable to directly characterize the degree of deviation. The ratio index is established as,
With the designed CPP surface in Fig. 3, Fig. 6 shows the simulation results of ratio index for the combination of any two kinds of DOFs. It was assumed that there was no processing error added in the designed CPP surface. Generally, the value of p rises rapidly with the misalignment of any DOF increases. If there only exists a translational deviation of 0.1 mm along one direction, the p value is approximately 3.3%. With translation along two directions, the value of p increases to 4.8%. It is interesting that the value of p followed the concentric distribution for translations, indicating that the influence along the two directions was the same, even though the CPP surface was not diagonally symmetric. Compared with translation, it seems that the rotation had a smaller effect on error analysis. When the only rotational deviation was 0.1 degree, the RMS was 1.9%. However, rotational misalignment cannot be neglected, because it is more likely to occur in the actual matching process.
4. Characterization method for imprinting CPPs
4.1 Principle of the automatic characterization method
Matching between the designed and measured CPP surfaces can be achieved manually, which depends heavily on the experience. However, this is unrepeatable and inefficient. Besides, the matching accuracy cannot be guaranteed. Thus, an automatic and efficient matching method needs to be developed for imprinting CPP topography.
For the iterative solution of the least squares problem established in Eq. (5), a reasonable initial guess is needed to avoid unexpected divergence and local minimum error [19]. Moreover, the precise alignment is also hard to achieve at one stage. Therefore, it is necessary to divide the whole matching procedure into two stages: coarse matching and fine matching. Coarse matching is used to search for an approximate area, and rapidly locate the measured data near the approximate position. Based on the initial position, fine matching is implemented to precisely align the measured data.
Above all, the flowchart of the proposed characterization method for CPP surface is illustrated as Fig. 7, which consists of three main stages: data pre-processing, surface matching and quality evaluation.
At the data pre-processing stage, the initial measured surface data from instruments are preprocessed with several necessary steps, because the measurement result cannot be directly matched with designed CPP surface. First, the outliers embedded into the measurement result significantly affect the matching process and need to be eliminated. Second, with edge truncation, the effective analyzed area is separated from the measurement result. Then, the transmitted wave front data are converted to the same unit with the designed surface. After that, based on the surrounding unprocessed area, the tilt is removed from the measurement result, because the starting measured surface is also needed. Finally, if necessary, smoothing is also needed to reject the noise of the data with the help of suitable filter technology.
Two-step matching is the critical part of the characterization method [31]. Considering the three DOFs, image registration method is adopted in the initial coarse matching. From this point, resampling with uniform and smaller data interval for the two matched surfaces is needed, which affects the matching accuracy. In conventional image registration, there is not much attention paid to the value difference of two image along the normal direction, because it essentially belongs to 2D registration. For the existence of processing error in height, the two images generated with the measured and designed surface is essentially different. In other word, the difference of two surfaces is out-of-plane, which makes the performance of image registration method limited. In this way, only approximate position is achieved. In the following fine matching, the nonlinear optimization is recommended to find the accurate solution by iteration.
With the output transformation matrices from matching process, the measured data are adjusted to acquire the form error for the quality evaluation, which also generate corresponding removal distribution in the next iteration step.
4.2 Coarse matching
After pre-processing, two datasets of designed and measured CPP surface can be transformed into two images, which are input into the coarse matching process, as shown in Fig. 8. With low computation cost and excellent robustness, the FFT-based method is adopted [32], which is the extension of the phase correlation method for automatic image registration. It can fully evaluate the matching performance from the aspects of translation, rotation, and scale, determining the optimal matching parameters in the Fourier domain.
This way, if ${f_1}({x,y} )$ and ${f_2}({x,y} )$ are the two surface datasets with the same grid interval that only differ in the translation of $({\Delta x,\Delta y} )$, ${f_1}({x,y} )$ can be expressed as,
The relationship between their Fourier transforms can be established as,
After that, the cross-power spectrum of two surfaces can be given by,
The inverse Fourier transform is further conducted, which generates an impulse. The two translation parameters can be separated according to the position of generated impulse. To recover rotation and scale, the phase correlation technique is extended by discovering their counterpart in the Fourier domain. As rotation $\Delta \theta$ and scale k are considered, the relationship between two surfaces and their corresponding Fourier transforms are also modified. They can be established as,
For the CPP matching problem, the scale deviation does not exist, so the scale k is determined to be 1. Suppose their corresponding magnitudes are ${M_1}({u,v} )$ and ${M_2}({u,v} )$, we can have,
It is worth noting that the translation disappears and the amplitude spectrum is only affected by the angular deviation of the two surfaces. Hence, their relationship in polar coordinates can be given by,
In polar coordinates, the angular deviation is transformed into an equivalent translation, which can be easily solved using the above phase correlation method. After the rotation is compensated, the phase correlation method is applied again to eliminate the real translation. At this point, both translation and rotation parameters are determined.
4.3 Fine matching
Nonlinear optimization is widely used in various fields, for which all the methods are iterative [33]. From the given point x0, a series of following points $\{{{\textbf{x}_1},{\textbf{x}_2},{\textbf{x}_3}, \cdots } \}$ are generated to decrease the objective function, which expectedly converges to ${\textbf{x}^ \ast }$ that corresponds with the minimizer of function. Nonlinear optimization focuses on how to effectively update x and tends to improve stability, reliability, and efficiency.
Generally, to minimize $f(\textbf{x} )$, the Steepest Descent method is adopted to update x by,
Although the best descending direction can be determined by the Steepest Descent method, the convergence gradually slows with the optimal approaching. Considering the good performance of coarse matching, the optimal value usually has been in a small range. In that case, the Gauss-Newton method suggests that,
However, ${\textbf{J}^\textbf{T}}\textbf{J}$ is not always positive definite, indicating that ${\textbf{p}_{gn}}$ may not be the descending direction. The Levenberg-Marquardt method combines the Steepest Descent method and the Gauss-Newton method. The iteration principle is optimized via a damping factor, and expressed as,
For $\lambda > 0$, the matrix ${\textbf{J}^\textbf{T}}\textbf{J} + \lambda \textbf{I}$ is positive definite, ensuring the validity of iteration direction. Besides, the rapid and stable convergence can be guaranteed by tuning the damping factor. Larger $\lambda$ can make the initial iteration stage perform well, especially when the current solution is far from the optimal. When it is close to the optimal, final quadratic convergence can be achieved a smaller $\lambda$.
Based on experience, the Levenberg-Marquardt method is preferred to determine the best parameters of transformation matrix. Note that, in each iteration step, the objective function is established with the proposed height tracing method, which is detailly discussed in the section 4.4. The detailed procedure of fine matching is presented below.
Step 1: Initialize the transformation matrix with the matching results from the coarse matching, and provide an initial damping factor.
Step 2: Calculate the value of objective function, and Jacobian matrix by differentiating the objective function.
Step 3: Update the transformation matrix with the Levenberg-Marquardt method, and obtain the new value of objective function.
Step 4: If the objective function decrease, keep the transformation matrix and reduce the damping factor. Otherwise, reject the transformation matrix and increase the damping factor.
Step 5: Repeat step 2 ∼ step 4 until the iteration number is reached. Output the final transformation matrix and precisely align the two surfaces.
4.4 Matching performance evaluation with the height difference tracing method
For two-step matching, Eq. (5) can be used to evaluate the matching performance for determining optimal matching. Considering the specific requirement of processing error, the RMS of height difference is directly adopted, which is equally minimized when the misalignment is eliminated. To implement this, the fiducial-aided height difference tracing method is proposed to evaluate the effectiveness of transformation matrix.
Suppose there are two misaligned surfaces, and they are discretized by a uniform grid (blue and black grid), as shown in Fig. 9. The datasets of the designed and measured CPP surface are represented by $\textbf{D} = \{{({{x_{1,i}},{y_{1,i}},{z_{1,i}}} )|{i = 1,2, \cdots } } \}$ and $\textbf{M} = \{{({{x_{2,i}},{y_{2,i}},{z_{2,i}}} )|{i = 1,2, \cdots } } \}$ respectively. Apparently, if there is no misalignment, two of the same smooth surfaces will have the same height value at any point. Therefore, one of the surface grids can be considered as the fiducial position for tracing height difference between the two surfaces. For example, the designed CPP surface is the reference. With three transformation parameters, the measured CPP surface data are adjusted to be $\textbf{M}^{\prime} = \{{({x{^{\prime}_{2,i}},y{^{\prime}_{2,i}},z{^{\prime}_{2,i}}} )|{i = 1,2, \cdots } } \}$, which can be determined as,
Based on this, the height value of adjusted measured surface at fiducial position $({{x_{1,i}},{y_{1,i}}} )$ is solved with the interpolation method. To reconstruct the smooth surface and get more accurate height values, the Clough-Tocher interpolation method is adopted [34], which is based on the triangulation of the given scattered points. In each triangle, the cubic interpolation scheme is carried out by using bivariate cubic polynomials. This way, C2-continuity is achieved to guarantee the smoothness of CPP surface. In Fig. 9, D1 is one of the fiducial positions, and its height value is constantly updated with its surrounding known points.
It is worth noting that there is not much attention paid to the search of corresponding points, which is directly achieved by the same fiducial positions. For convenience, $({{x_{2,i}},{y_{2,i}}} )$ is set to be equal to $({{x_{1,i}},{y_{1,i}}} )$.
5. Verification
5.1 Matching accuracy
To evaluate the matching accuracy of the proposed characterization method, the numerical simulation was conducted with the pure designed CPP surface. An arbitrary transformation matrix (Tx = 10 mm, Ty = 5 mm and rz = $- 1$°) was introduced to generate the measured CPP surface, as shown in Fig. 10. The whole matching process is illustrated in Fig. 11. With misalignment, the measured CPP surface could not be directly located at the exact area of designed CPP surface, as shown in Fig. 11(a). Compared with the theoretical zero error, the form error evaluation could not be conducted, which introduced significantly false analysis results. After coarse matching, the measured CPP surface was approximately located near the correct area as shown in Fig. 11(b). Figure 11(d) indicates that there was residual misalignment, and the residual error was RMS 60.36 nm. In the fine matching, the slight adjustment was implemented as shown in Fig. 11(c). According to Fig. 11(e), the error was further reduced to RMS 0.2 nm, which demonstrates that the sub-nanometer accuracy is achieved.
Fig. 11. Matching process. (a) Before matching; (b) After coarse matching; (c) After fine matching; (d) Residual error after coarse matching; (e) Residual error after fine matching.
The RMS evolution of residual error in fine matching is shown in Fig. 12. After coarse matching, the calculated results ${\textbf{x}_0} = ({ - \textrm{10}\textrm{.0149, } - \textrm{5}\textrm{.1099, 0}\textrm{.9114}} )$ were close to the potential true value, which controlled the matching error in a small range. In the following iteration of fine matching, the residual error continuously decreased and rapidly converged to a stable value, which proved the superiority of Levenberg-Marquardt method. The optimal transformation parameters were finally determined to be ${\textbf{x}^ \ast } = ({ - \textrm{9}\textrm{.9993, } - \textrm{5}\textrm{.0008, 1}\textrm{.0006}} )$. Therefore, the matching accuracy was 0.0001 mm in translation and 0.0001°in rotation, indicating the high precision of the proposed characterization method. Above all, it can meet the demand of form error evaluation for imprinting CPPs.
5.2 Performance of form error separation
For the CPPs matching problem, another main task of matching is to generate the corresponding removal distribution for the figuring process. To investigate this, the measured CPP surface was obtained with the artificial processing error distribution added on the designed surface, and the analyzed area was same as that in Section 5.1. The whole process is simplified by starting with the fine matching. The measured CPP surface was slightly adjusted with the transformation matrix (Tx = $- 0.1$ mm, Ty = 0.1 mm and rz = 0.08°) to simulate the matching result from coarse matching. The two-dimensional sinusoidal waves are defined as the error to be added, and the expression was,
where A is the amplitude, and T is the spatial wavelength. The RMS of error distribution was set to be 30 nm (A = 60 nm and T = 30 mm), and the matching results are shown as Fig. 13.Fig. 13. Artificial error and matching results. (a) Artificial error distribution; (b) Before fine matching; (c) After fine matching
After fine matching, the corresponding designed CPP surface was subtracted from the adjusted measured CPP surface to separate the form error. Figure 14 shows the comparison between the true and separated form errors. From the matching result, the form error was successively separated from the measured CPP surface. As shown in Fig. 15, the RMS of initial unmatched form error was 82.08 nm, indicating the misalignment effect on form error evaluation. With high fidelity, the RMS of separated form error was 29.76 nm, which was quite close to the theoretical value 30 nm and indicates the strong analysis ability of the proposed characterization method.
Fig. 14. Comparison of true and separated form error. (a) True form error; (b) Separated form error.
5.3 Experimental verification
To further study the capability and performance of the proposed method in the experiment, the CPP surface was fabricated by APPP [35,36]. As an innovative deterministic processing technology, APPP has great potential for imprinting complex CPP topography with higher efficiency and a lower cost. The experiment consisted of two processing stages. First, the CPP topography was imprinted on the flat surface. Then, with the proposed characterization method, the form error evaluation was conducted, which was figured in the next iteration. Considering the complexity of CPPs, the dwell time map was calculated with modified convolution iteration method. Compared with the method by solving linear equations in algebra, it can be achieved with lower calculation cost and better smoothness.
Figure 16 shows the matching results of the first processing stage. According to the measured surface, the corresponding area (RMS 198.8 nm) was searched in the whole designed CPP surface. After two-step matching, the form error was determined to be RMS 62.4 nm, indicating that the convergence rate of RMS was nearly 70%. There were mainly two errors during APPP process: the nonlinearity and stability of the TIFs. It was difficult to completely compensate the nonlinearity of TIF, and the form error was largely embodied in the height deviation, which indicates a strong connection with the designed CPP features.
Fig. 16. Matching results of the first processing stage. (a) Designed surface; (b) Measured surface after matching; (c) Form error distribution.
Figure 17 shows the matching results of the second processing stage. It should be noted that the form error distribution was consistent with that in the first processing stage. As shown in Fig. 18, the power spectrum density (PSD) analysis of form error was conducted. Limited by the size of TIF, only the form error within the range of f < 0.3768 mm−1 was successfully figured in this stage, which indicates the advantages of the APPP process. Finally, the form error was RMS 43.7 nm. Above all, the proposed characterization method can be used to evaluate the processing quality and guide the iteration.
Fig. 17. Matching results of the second processing stage. (a) Measured surface after matching; (c) Form error distribution.
6. Conclusions
In this paper, the misalignment effect and the characterization method for imprinting CPPs were investigated, which serves for form error evaluation and removal generation in the iteration processing. From the simulated and experimental results, the following main conclusions were drawn as followed:
- (1) With the essence of flat figuring and least-squares plane fitting, the matching DOFs for CPP surface are simplified into three DOFs (Tx, Ty, rz). Based on this, the mathematic model is established, which is a typical least squares problem.
- (2) Form error evaluation of the CPP surface is susceptible to the misalignment effect, leading to misjudgment in iteration. Considering the CPP features, the form error deviation caused by the misalignment is mainly attributed to the surface components with lower frequency. It is worth noting that the deviation is more sensitive to translation misalignment compared with rotation misalignment.
- (3) To improve the efficiency and accuracy, an automatic characterization method based on the two-step strategy and height tracing method is proposed. Image registration and nonlinear optimization are utilized for the coarse and fine matching. The matching performance can be effectively evaluated by the proposed height tracing method.
- (4) Based on the simulations, the proposed characterization method achieves sub-nanometer accuracy for imprinting CPPs. The form error can be separated from the measured surface with high fidelity, indicating that the reasonability and feasibility of proposed method. The verification experiments imply that the characterization method can be applied successfully in the fabrication of CPPs.
Appendix. Nomenclature
$T$ | Transformation matrix |
${P_i}$ | The ith point on the designed surface |
${Q_i}$ | The ith point on the measured surface |
${T_x},{T_y},{T_z}$ | Translation parameters in transformation matrix |
${r_x},{r_y},{r_z}$ | Rotation parameters in transformation matrix |
$z$ | Assumed smooth surface |
$\Delta z$ | Height changes caused by misalignment |
${z_{1,i}},{z_{2,i}}$ | Height of the ith point of the designed and measured surfaces |
$\Delta {z_i}$ | Height compensation for the ith point after eliminating the misalignment |
$N$ | Number of point pairs |
${S_0}$ | Designed CPP surface |
${E_0}$ | True form error |
$E$ | Analyzed form error |
$\Delta E$ | Form error deviation |
$p$ | Ratio index to characterize the degree of deviation |
$T{M_c}$ | Transformation matrix obtained by coarse matching |
$T{M_f}$ | Transformation matrix obtained by fine matching |
${f_1},{f_2}$ | Surface dataset |
${F_1},{F_2}$ | Fourier transform |
${C_{cor}}$ | Cross-power spectrum |
${M_1},{M_2}$ | Amplitude spectrum |
$\Delta x,\Delta y$ | Translation |
$\Delta \theta$ | Rotation |
$k$ | Scale |
${\textbf{x}_0}$ | Initial point of nonlinear optimization |
${\textbf{x}^\ast }$ | Optimal solution of nonlinear optimization |
${\textbf{p}_{sd}}$ | Increasement with the Steepest Descent method |
${\textbf{p}_{gn}}$ | Increasement with the Gauss-Newton method |
${\textbf{p}_{lm}}$ | Increasement with the Levenberg-Marquardt method |
$\textbf{J}$ | Jacobian matrix |
$\lambda$ | Damping factor |
$\textbf{D}$ | Dataset of designed surface |
$\textbf{M}$ | Dataset of measured surface |
$\textbf{M}^{\prime}$ | Adjusted dataset of measured surface |
CPPs | Continuous phase plates |
CCOS | Computer controlled optical surfacing |
MRF | Magnetorheological finishing |
IBF | Ion beam figuring |
BP | Bonnet polishing |
APPP | Atmospheric pressure plasma processing |
CRDM | Coupled reference data method |
SRS | Structured region signature |
AFEEM | Automatic form error evaluation method |
AICP | Adaptive iterative closest point |
IFPAM | Intrinsic feature-based pattern analysis method |
IFPFC | Invariant feature-pattern-based form characterization |
TIF | Tool influence function |
DOFs | Degrees of freedom |
ICP | Iterative closest point |
RMS | Root mean square |
PSD | Power spectrum density |
Funding
Heilongjiang Provincial Natural Science Foundation of China (LH2020E039); National Natural Science Foundation of China (51905130).
Acknowledgments
The authors would like to sincerely thank the reviewers for their valuable comments on this work.
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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