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Abstract
The surface figure accuracy requirement of cylindrical surfaces widely used in rotors of gyroscope, spindles of ultra-precision machine tools and high-energy laser systems is nearly 0.1 µm. Cylindricity measuring instrument that obtains 1-D profile result cannot be utilized for deterministic figuring methods. Interferometric stitching test for cylindrical surfaces utilizes a CGH of which the system error will accumulated to unacceptable extent for large aperture/angular aperture that require many subapertures. To this end, a self-calibration interferometric stitching method for cylindrical surfaces is proposed. The mathematical model of cylindrical surface figure and the completeness condition of self-calibration stitching test of cylindrical surfaces were analyzed theoretically. The effects of shear/stitching motion error and the subapertures lattice on the self-calibration test results were analyzed. Further, a self-calibration interferometric stitching algorithm that can theoretically recover all the necessary components of the system error for testing cylindrical surfaces was proposed. Simulations and experiments on a shaft were conducted to validate the feasibility.
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1. Introduction
Cylindrical surfaces are increasingly used in spindles of ultra-precision machine tools and high-energy laser systems [1–3], etc. As the performance of these systems enhances, the surface figure accuracy (roundness/cylindricity for shaft surfaces) requirements on cylindrical surfaces are enhanced. The roundness of the spindle of the state-of-the-art ultra-precision lathe - Nanoform 700 ultra should be better than 0.1 µm to achieve 15 nm rotation precision [4]. The surface figure error of typical cylindrical silicon mirror in resonators [1,2] and beam shaping units [3] of high-energy laser systems are required to be better than 0.1 µm peak-to-valley (PV) to meet the energy concentration requirement. For achieving the required form accuracy efficiently, deterministic figuring process of high-energy cylindrical mirrors has been already applied [5] and deterministic figuring methods for shaft parts were recently proposed by our research team [6]. As the premise of deterministic figuring, surface figure test of the cylindrical surfaces is essential. Commercial cylindricity measuring instrument, e.g., Talyrond series can be used for measuring roundness and cylindricity [7]. However, the sample resolution of the test result along the axis direction of the test surface is not sufficient to direct the deterministic figuring process. Moreover, the contact probe may scratch the test surface. This is not allowable for surfaces with low hardness materials e.g., aluminum, and low damage surfaces applied in high-power and high-energy laser applications. As the commonly used non-contact test method with high accuracy and resolution, wavefront interferometric test method usually requires a customized computer-generated-hologram (CGH) to transform the flat wavefront into cylindrical wavefront [8,9]. Test accuracy of the CGH method depends on the accuracy of the CGH, which mainly comprises the CGH pattern error (caused by duty-cycle error, etching depth error, and pattern distortion [10]), substrate error (caused by surface figure of the two surfaces of the substrate and the refractive index inhomogeneity). Commonly, a faster CGH (i.e., a CGH with smaller F/#) has larger pattern error due to the smaller grating period. The substrate error grows with the dimension of the CGH [11].
Limited to state-of-the-art manufacture process of CGHs, commercial CGH cylinder nulls usually have limited dimensions (as large as Φ 6”) and F/# (as fast as F/1). Therefore, stitching along the arc direction and/or the axial direction is required to obtain the full aperture test result [12–17]. On the other hand, to reduce the required number of subapertures, a faster and larger CGH is required for testing shaft surfaces and cylindrical mirrors with large aperture. For a typical CGH with F/2 and 120 mm × 120 mm aperture (model No. H120F2C, produced by DiffractionTM International), the pattern error and substrate error is 0.12 fringes and 0.4 fringes PV, respectively [11]. Therefore, the CGH error cannot be neglected for testing cylindrical surface with accuracy requirement better than 0.1 µm. Further, the CGH error will exist in measurement results of all subapertures, and it will be accumulated and enlarged in the stitching result. Therefore, calibration of the CGH error is essential for testing cylindrical surface, especially for shaft surfaces or cylindrical mirrors with large aperture that requires a CGH with a fast F/# (e.g. faster than F/2) and large aperture (e.g. bigger than 100 mm).
The existing methods to troubleshoot the calibration issue can be categorized as the substrate calibration method and the absolute test method. Ping Zhou reported two substrate calibration methods for aspheres CGH nulls [10]. The in-situ measurement method tests the 0-order transmissive wavefront distortion of the CGH with a high accuracy return sphere replacing aspheric surface under test. The other method simply tests the 0-order transmissive wavefront distortion in the collimated beam. However, pattern error of the CGH cannot be calibrated by this method. Further, the 1-order test beam does not transmit along the same path as the beam in the calibration configuration. The difference will be worse when the etching area is on the side of CGH that faced to the interferometer and the CGH has a fast F/#. Theoretically, substrate calibration method is an approximate method for calibrating CGH error, and it can roughly calibrate CGHs with relatively large F/# [10] (e.g., F/6 with neglectable PV 0.04 fringes pattern error). In terms of the absolute method for cylinder CGH nulls, Ma et al. reported a conjugate differential method. The method introduces a certain shift of the test cylindrical surface both parallel to and rotated about the centerline [18] and use the differential maps of CGH error to reconstruct the CGH error. Reardon et al. reported a “random fiber” test method using a high accuracy fiber to perform the cylindrical surface version of random ball test [19]. These absolute methods require an additional and sophisticated test system. Further, there exists registration error when subtracting the calibrated CGH error from the test result. This registration error will accumulate in stitching test for shaft surfaces or cylindrical surfaces with a large aperture that require many subapertures to cover the full aperture of the test surface.
Self-calibration technique has already been used in stitching test of large flats [20] and spheres [20,21] to estimate the repeatable reference errors. The merit is that the calibration process is performed simultaneously with the stitching process. The calibration configuration is in situ; hence no registration error exists when subtracting the calibrated reference error from the test results. However, certain surface figure components of the reference surface will lose during the self-calibration [22,23]. Theoretical analysis shows that the 2nd terms i.e., x2, y2, and xy terms of the reference error cannot be correctly determined by any self-calibration algorithms due to the existence of the misalignment aberrations in flat tests. Therefore, the 2nd terms must be removed from the stitched surface figure of the full aperture. The research on self-calibration stitching test of flats and spheres paves the way for its application on cylindrical surfaces. However, the influence of cylindrical surfaces misalignment aberrations on the completeness of the self-calibration result of the CGH error and the stitching result remains unclear. Further, the influence of the subapertures partition strategy on the self-calibration result and the stitching results is ambiguous. The insufficient research on the self-calibration stitching test method of cylindrical surfaces hinders high accuracy interferometric test of shaft surfaces and cylindrical surfaces with large aperture to some extent. Thus, the development of a self-calibration interferometric stitching method for cylindrical surfaces, whereby sufficient information of the cylinder CGH nulls error for testing cylindrical surfaces can be obtained, and thorough analyses of the completeness condition of self-calibration stitching test for cylindrical surfaces remain challenging.
To this end, we present a self-calibration interferometric stitching method for cylindrical surfaces. The completeness condition for obtaining the sufficient information to reconstruct the stitched surface figure free of CGH null errors is derived. Subaperture partition strategy to avoid the loss of period error and certain terms error; and the strategy to avoid misalignment-induced uncertainties are presented (Section 2). Simulations show that the surface figure of cylindrical surfaces can be stitched free of any additive CGH error (Section 3). A shaft surface with a radius of 30 mm and height of 120mm was tested to validate the feasibility of the proposed method (Section 4). Discussions are included in Section 4. Finally, the paper is concluded in Section 5.
2. Principle
2.1 Cylindrical surfaces surface figure representation model
Wavefront interferometric test is a relative test method that compares the surface under test with the shape of the wavefront incident to the test surface. For testing cylindrical surfaces or shaft surfaces, the reference wavefront is cylindrical wavefront. Therefore, establishing a general model for representing surface figure of cylindrical surfaces is essential for developing the self-calibration stitching method. Suppose a point on a cylindrical surface can be expressed as cylindrical coordinates (θ,z, ρ) as shown in Fig. 1. The cross section of a shaft can be expanded as Fourier series as
Shaft surface is a continuous surface with circular cross section. By replacing ak and bk with functions of axial coordinate z, i.e., ak(z) and bk(z), the shaft surfaces can be expanded as
Substitute Eq. (3) into Eq. (2), yields
Therefore, the shaft/cylindrical surfaces figure error with no basic axis position requirements can be expressed as
Table 1. The first 16 terms of LF polynomials.
2.2 Completeness condition for self-calibrate system error of cylindrical surfaces testing and the herein algorithm
Self-calibration stitching algorithm is a kind of stitching algorithms. Therefore, analyzing the traditional algorithm for stitching test of cylindrical surfaces is a good start. The key of stitching algorithms is to balance misalignment aberrations of different subapertures to make the surface figure of the common area have the same phase amplitude. For interferometric null test of shaft or cylindrical surfaces, cosθ, sinθ, zcosθ, and zsinθ (i.e., the second and the third term in Eq. (4) represents the misalignment aberration introduced by translation along x and y, tilt about x and y, respectively [8]. The directions of x and y are shown in Fig. 1. Therefore, these terms are utilized as free-compensators for each subapertures to stitch all subapertures by minimizing the difference within the overlapping region of all subapertures [26]. The common stitching algorithm for cylindrical surfaces can be expressed as
To realize self-calibration of system error during the stitching process, it is important to understand the essence of the self-calibration theory. Self-calibration theory is similar with wavefront shear reconstruction theory that utilize differential information to recover the system error [27–33]. The general shear reconstruction/self-calibration model with shear amounts (translation amount between subapertures) along two orthogonal directions as shown in Fig. 2 can be expressed as [34,35]
Fig. 2. The general shear reconstruction/self-calibration model with shear amounts (translation amount between subapertures) along two orthogonal directions.
The system error R(x, y) can be reconstructed from its difference wavefronts along x and y. If there exists no shear motion error (i.e., translation motion error), the completeness condition is that there are at least two shear amounts $s_{{j_1},{k_1}}^x$ $s_{{j_2},{k_2}}^x$ along x axis and two shear amounts $s_{{j_1},{k_1}}^y$ $s_{{j_2},{k_2}}^y$ along y axis. Moreover, the greatest common divisor of $s_{{j_1},{k_1}}^x$ and $s_{{j_2},{k_2}}^x$, $s_{{j_1},{k_1}}^y$ and $s_{{j_2},{k_2}}^y$ should be 1 [33,35, 36]. The completeness condition for shear reconstruction from the shear maps was discussed in frequency- domain [33] and space domain [35, 36] respectively. The strategy how to choose the shear amounts are analyzed in Ref. [35].
However, shear motion error exists in practice. It will inevitably introduce misalignment errors into the measurement results. The misalignment errors will result in unambiguous reconstruction of certain terms of the system error R. For testing flat surfaces, the misalignment aberration is linear combination of ax, by, and c. Its integral over x is linear combination of ax2, bxy, cx, and f(y). Considering the system error R(x, y) contains ax2, bxy, cx, and f(y). ax2, bxy, cx, and f(y) has to be recovered from ΔR(x, y). The difference value of ax2, bxy, cx, and f(y) with respect of x is ax2−a(x−Δx)2 = 2aΔx·x−a(Δx)2, bxy−b(x−Δx)y = bΔx·y, cx−c(x−Δx) = cΔx, f(y)−f(y) = 0. This means that the difference value of ax2, bxy, cx, and f(y) with respect of x are mixed with the misalignment aberration, therefore the terms of ax2, bxy, cx, and f(y) of the system error cannot be recovered from the difference value along x. Similarly, integrate the misalignment aberrations over y. axy, by2, cy, and f(x) can be obtained. The difference value of axy, by2, cy, and f(x) with respect of y is axy−ax(y−Δy) = aΔy·x, by2−b(y−Δy)2 = 2bΔy·y− b(Δy)2, cy−c(y−Δy) = cΔy, f(x)−f(x) = 0. These terms are mixed with the misalignment aberration, therefore the terms of axy, by2, cy, and f(x) of the system error cannot be recovered from the difference value along y. The intersection set of {ax2, bxy, cx, f(y)} and {axy, by2, cy, f(x)} is the terms that cannot be recovered from the difference map along both x and y, i.e., {ax2, bxy, cy2, dx, ey, f}. The linear combination of dx, ey, f is also a flat surface. Therefore, the second-order term ax2, bxy, cy2 cannot be recovered by the shear reconstruction method with difference wavefronts along x and y. In other words, the self-calibration stitching method with subapertures arranged along x and y axes cannot recover ax2, bxy, and cy2 unless the tilt error and roll error of the stitching translation are monitored by extra instrument.
Following the above analysis method for determining the terms that cannot be reconstructed from the self-calibration stitching test of flat surfaces, the case for cylindrical surfaces were analyzed. For testing cylindrical surfaces, the misalignment aberrations is linear combinations of sinθ, cosθ, zsinθ, and zcosθ. Its integral over θ is linear combination of cosθ, sinθ, zcosθ, zsinθ, and f(z). Therefore the terms cosθ, sinθ, zcosθ, zsinθ, and f(z) of the system error cannot be recovered from the difference value along θ. Integrate misalignment aberrations over z generate combination of zcosθ, zsinθ, z2cosθ, z2sinθ, and f(θ). Therefore the terms zcosθ, zsinθ, z2cosθ, z2sinθ, and f(θ) of the system error cannot be recovered from the difference value along z. The intersection set of {cosθ, sinθ, zcosθ, zsinθ, f(z)} and {zcosθ, zsinθ, z2cosθ, z2sinθ, f(θ)} is the terms that cannot be recovered from the difference value along both θ and z, i.e., {cosθ, sinθ, zcosθ, zsinθ, a}. cosθ, sinθ, zcosθ, and zsinθ represent the misalignment aberrations. a represents the radius error. Therefore, the surface figure error of the system error can all be reconstructed for a cylindrical surface with no basic axis position and radius error requirements.
Therefore, the completeness condition for recovering system error of cylindrical surfaces figure testing system is shears exist along both θ and z axes. Moreover, there are at least two shear amounts $s_{{j_1},{k_1}}^\theta$ $s_{{j_2},{k_2}}^\theta$ along θ axis and two shear amounts $s_{{j_1},{k_1}}^z$ $s_{{j_2},{k_2}}^z$ along z axis. The greatest common divisor of $s_{{j_1},{k_1}}^\theta$ and $s_{{j_2},{k_2}}^\theta$, $s_{{j_1},{k_1}}^z$ and $s_{{j_2},{k_2}}^z$ should be 1.
The self-calibration stitching algorithm can be pixel-relation-based algorithms or the orthogonal polynomials-relation-based algorithms [37]. The pixel-relation-based algorithms commonly exceed the pixel-relation-based ones on accuracy. However, the pixel-relation-based algorithms demands unacceptable computation time. Combining the representation of cylindrical surface figure, the orthogonal polynomials-relation-based self-calibration stitching algorithms can be expressed as
After the optimization, surface figure of all subapertures eliminating the reference error and misalignment aberrations can be obtained. Self-calibrated interferometric stitching test results can be obtained.
3. Simulations
The cylindrical surface in the simulation is a shaft with radius of 30 mm and length of 100 mm. A CGH with limited aperture is used to test the shaft. The test region on the shaft is 60° along the arc direction and 60 mm along the axial direction. The surface figure error of the shaft and the system error of the cylindrical wavefront generated by the interferometer and the CGH are simulated by a linear combination of LF polynomials. The coefficients of the LF polynomials for the surface figure error of the shaft are p = [0; 0; 0.1; 0; 0.1; 0; 0.1; 0; 0.1; 0.1; 0.1; 0; 0.1; 0; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1; 0.1]·λ. p_r = 0.02·ones(16,1)·λ for the system error, where λ = 632.8nm and ones(16,1) denotes a vector with 16 elements that are all 1. The surface figure error of the shaft and the system error are shown in Figs. 3 and 4 with PV 1027.57 nm, and 178.69 nm, respectively.
To test the full aperture of this shaft, three rows of subapertures are arranged. The coordinates of subapertures are listed in Table 2. The subaperture partition meet the completeness condition for recovering system error of testing cylindrical surfaces. Two shear amounts (i.e., 23° and 29°) along θ axis and two shear amounts (i.e., 21mm and 26mm) along z axis. The greatest common divisor of 23° & 29°, and 21mm & 26mm are 1. The simulated test result of subapertures 14, 26, 1, 13, 27, and 39 are shown in Figs. 5(a), 5(b), 5(c), 5(d), 5(e), and 5(f) respectively.
Fig. 5. The simulated test result of (a) subaperture 14, (b) subaperture 26, (c) subaperture 1, (d) subaperture 13, (e) subaperture 27, and (f) subaperture 39.
Table 2. The coordinates of subapertures.
The stitching result without calibrating the system errors is shown in Fig. 6(a). After subtracting the nominal full aperture surface figure, the stitching error can be obtained as shown in Fig. 6(b) with 315.34 nm PV and 85.38 nm RMS. It shows that the distribution of the stitching error is similar with that of the system error, however, the error accumulated to an unacceptable amount. Therefore, calibration of the system error is necessary to obtain higher test accuracy. Then the self-calibration stitching test algorithm as shown in Eqs. (11–13) was utilized to stitch the subapertures. The stitching result is shown in Fig. 7(a). After subtracting the nominal full aperture surface figure, the stitching error can be obtained as shown in Fig. 7(b) with 0.09 nm PV and 0.01 nm RMS, which is neglectable. The recovered system error is shown in Fig. 8(a). After subtracting the nominal system error, the system error calibration error can be obtained as shown in Fig. 8(b)with 0.02 nm PV and RMS value smaller than 0.01 nm, which is also neglectable.
Fig. 7. The stitching result (a) and the stitching error (b) utilizing the self-calibration stitching algorithm.
Fig. 8. The recovered system error result (a) and reconstruction error (b) utilizing the self-calibration stitching algorithm.
4. Experiments
Experiments were conducted to verify the proposed method. the cylindrical surface under test is a shaft with radius of 30 mm and height of 100 mm as shown in Fig. 9. A synchronous phase shift interferometer Zygo GPI 6” is utilized. A stitching workstation is developed for testing cylindrical surfaces. To accommodate the heigh of the stitching workstation [16], two steel frames are fabricated to support the interferometer and the CGH, respectively. The arrangement is easily to be affected by vibration. Therefore, an air bearing table is utilized to reduce the vibration effect. Moreover, the test is performed at midnight with all fabrication machine off. The displacement accuracy of all the three linear axes is 30 µm. The range of the rotary stage is 360°; the motion resolution and repeated positioning accuracy are 0.00125° and 0.005°, respectively.
A f/2 CGH is designed to test the shaft. The optical design process of the CGH can be referred to ref. [16]. The optical layout for designing the CGH is shown as Fig. 10(a). The distance between the CGH back surface to the shaft is 90 mm. The test region of the CGH is 63 mm × 63 mm. The test region on the shaft is about 63 mm × 28° (linear direction× arc direction). Figure 10(b) shows the residual deigned aberration with PV 7.06×10−7λ (λ = 632.8 nm). The design accuracy meets the test accuracy requirements. The contour map of the CGH phase within the test region is shown in Fig. 10(c). The contour spacing is 500 periods. Figure 10(c) indicates that the average spacing is estimated to be 63 mm/32/500 = 4 µm. The phase defined by Zernike polynomials within the test and alignment regions of the CGH was then approximated by line segments on the CGH substrate with preset tolerance. These line patterns were encoded to GDSII format [38] shown as Fig. 11(a) for fabrication. The fabricated CGH is as shown in Fig. 11(b).
Fig. 10. Optical design of the CGH. (a) The optical layout, (b) the designed residual error, and (c) the contour map of the CGH phase within the test region.
The subapertures lattice shown in Fig. 12 meets the completeness condition for recovering system error of cylindrical surfaces figure testing system. 51 subapertures with shears along both θ and z axes are arranged to cover the full apertures. There exist three rows subapertures with shears of 29mm and 37mm along z. The greatest common divisor between 29mm and 37mm is 1. Each row has 17 subapertures. The first 9 subapertures of each row have angle interval of 23°, and 19° for the last 8 subapertures. The greatest common divisor between 23° and 19° is 1.
The stitching test scheme for this shaft requires translational motions along the axis of the shaft (z-axis) and rotational motion around the central axis of the shaft (C-axis). Further, translation and rotation (A-axis) along/around the optical axis (x-axis) to remove power and twist, translation and rotation (B-axis) along/around y-axis to remove x-tilt and y-tilt are necessary for aligning the test shaft to the CGH. Therefore, the stitching motion system requires a 2-axis (z and C) electric control motion system and a 4-axis (x, y, A and B) manual (or electric controlling) adjusting table.
As shown in Fig. 13, a stitching motion system was established based on the requirements of movement. The motion system comprises four electric control motion axes (x, y, z, and C) and a two-axis manual adjusting table (A and B). The stokes of z-axis is 700 mm. The positioning accuracy for each translation axis is approximately 30 µm. The range of the rotary stage (C) is 360°; the motion resolution and repeated positioning accuracy are 0.00125° and 0.005°, respectively. A set of Zygo GPI 6” is utilized as the interferometer, and the CGH is located between the interferometer and the test surface.
After obtaining all subapertures test results, Eqs. (11–13) was utilized to stitch the subapertures. The self-calibration stitching test result is shown in Fig. 14 with 766.83 nm PV and 76.26 nm RMS. The recovered system error is shown in Fig. 15 with 44.7 nm PV and 5.12 nm RMS. For comparisons, the stitching result without calibrating the system errors is shown in Fig. 16 with 599.22 nm PV and 72.65 nm RMS. Subtracting the self-calibration stitching test result as shown in Fig. 15 from Fig. 16, periodic error as shown in Fig. 17 with 152.64 nm PV and 14.38 nm RMS appears. It can be explained as effects of the system error on the test result. Therefore, the proposed method can enhance the test accuracy by eliminating the effects of system errors on the test result.
Fig. 17. The result of subtracting the self-calibration stitching test result as shown in Fig. 15 from Fig. 16.
As a secondary proof to the accuracy of the system error calibration result, the system error was tested by measuring the wavefront distortion using the zero-order diffraction from the CGH. The calibration system was set up as shown in Fig. 18. A retro-flat with very high accuracy is placed after the CGH. The test result is shown as Fig. 19 with 49.48 nm PV and 5.37 nm RMS. The phase distribution and PV/RMS value of the system error shown in Figs. 15 and 19 are very similar, which verified the proposed method to some extent.
5. Discussion
When compared with state-of-the-art absolute test methods for testing cylindrical surfaces, the proposed method has merits that the system error calibration is conducted simultaneously with the stitching process. No additional process and equipment are required besides the stitching platform. The calibration is performed in situ, therefore no registration error exists.
Although the self-calibration stitching method has been proposed for testing flat and spherical surfaces, applying the self-calibration method for cylindrical surfaces test still has many issues. One is the appropriate representation of the system error. Zernike polynomials can be utilized to describe system errors for testing flat and spherical surfaces. However, it is not applicable for cylindrical surfaces especially for shaft surface. Legendre polynomials can represent cylindrical surfaces with rectangle aperture. However, it is not suitable for 360° cylindrical surfaces, i.e., shafts. We theoretically analyzed the surface figure of cylindrical surfaces using a combination of Fourier polynomials and Legendre polynomials to solve this issue.
Further, the completeness condition for recovering the system error of cylindrical surfaces using the self-calibration method remains unknown. We have theoretically analyzed the influence of the shear/stitching motion on the system error reconstruction result, and concluded that the self-calibration method can recover all the necessary components of system error for testing cylindrical surfaces with no basic axis position requirements. Notably, using the self-calibration method for testing flat surfaces is not the case. The second order terms, i.e., x2, y2, xy terms cannot be recovered using the traditional self-calibration method that perform flat surfaces stitching along two orthogonal directions.
As analyzed in the principle part, the self-calibration interferometric test is similar with the shear reconstruct and absolute test using shift motions. Therefore, the accuracy of the test method is restricted by the temperature fluctuation & inhomogeneity, motion accuracy etc. The proposed method mainly focuses on the theoretical algorithm and the completeness conditions. The influence of the environment and equipment is complex and tedious that require specific analysis. We are working on it and hope to report it soon.
6. Conclusion
The mathematical model of cylindrical surface figure and the completeness condition of self-calibration stitching test of cylindrical surfaces were analyzed theoretically. Cylindrical surfaces with higher accuracy requirement that beyond the CGH accuracy can be test with system error calibrated using the proposed algorithm. The proposed method advances the stitching mode for cylindrical surfaces from relative test mode to absolute test mode that can simultaneously reconstruct the system error. A shaft with radius of 30 mm and height of 100 mm was successfully tested with system error simultaneously recovered and separated. The proposed method will help fabricate shaft with accuracy better than 0.1 µm to make spindles with nano-scale accuracy for ultra-precision machine tools. The proposed method can also help enhance the output energy of high energy laser system that require high accuracy large cylindrical surfaces.
Funding
National Natural Science Foundation of China (52105567, 51991371, 51835013); National Key Research and Development Program of China (2019YF0708903).
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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