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Subaperture stitching test for large aperture mild acylinders
Optics Communications 455 (2020) 124526
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Subaperture stitching test for large aperture mild acylinders Lingwei Kong, Shanyong Chen ∗ College of Artificial Intelligence, National University of Defense Technology, Changsha, Hunan, 410073, China Hunan Key Laboratory of Ultra-Precision Machining Technology, Changsha, Hunan, 410073, China Laboratory of Science and Technology on Integrated Logistics Support, National University of Defense Technology, Changsha, Hunan, 410073, China
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Keywords: Large aperture acylinder Subaperture stitching Near-null Surface reconstruction
ABSTRACT Large aperture acylinders are typical optical elements in synchrotron radiation systems and laser shaping devices. The manufacture of such ultra-precision optical elements requires ultra-precision measurement of the surface figure. The use of computer-generated hologram (CGH) customized for the acylindrical surface can only achieve a ‘‘one-to-one’’ test. Limited by the aperture of CGH, it also cannot meet the requirements for large aperture acylinders. In order to realize flexible test of large aperture acylinders with mild aspherical departure, a method of surface figure reconstruction for subaperture measurements under near-null test is proposed. It is based on ray tracing and rigid body transformation. By using a general CGH for cylindrical surface, near-null test for various acylindrical surfaces can be realized. Subaperture stitching test can overcome the problem of the limited CGH aperture which cannot meet the requirement of large aperture acylinders. Finally an elliptical cylindrical mirror with an aperture of 240mm (sagittal) × 350mm (tangential) is measured. And the cross test is realized by using a customized CGH to verify the validity and precision of the proposed method.
1. Introduction Mild acylinder refers to the aspherical cylinder whose generatrix is close to the circle, i.e. with mild aspherical departure. Similar to the cylindrical surface, mild acylinder has different curvatures in tangential and sagittal directions, respectively. It is often used for correcting astigmatic aberration, line focusing, and line imaging. Compared with the cylinder, acylinder introduces conic constants and even high-order terms, which indicates more degrees of freedom for optimization design and better performance can be achieved. So it is more and more widely used in synchrotron radiation systems and laser shaping devices. For example, the KB (Kirkpatrick–Baez) mirror system [1] which is composed of a pair of acylindrical mirrors has been widely used due to its advantages of correcting astigmatism, simple structure, and relatively low processing and assembly requirements [2–4]. For the first time, Suzuki et al. constructed a KB mirror system using mutually perpendicular elliptical cylindrical reflectors [5], which can better eliminate the spherical aberration compared to the KB system constructed by conventional cylindrical surfaces. The KB mirror system of PEEM in Shanghai synchrotron radiation facility [6] uses two ultra-precision elliptical cylindrical mirrors with aperture of 250 mm (sagittal) × 40 mm (tangential), 450 mm (sagittal) × 30 mm (tangential), and the surface slope error is better than 5 μrad. The optical configuration of focusing system of SPring-8 angstrom compact free-electron laser (SACLA) adopts two-stage KB mirror system [7], which uses a total of four elliptical cylindrical reflectors. Recently, the European X-ray Free
Electron Laser (XFEL) has successfully produced its first X-ray photon pulse trains. A KB mirror pair is designed to focus hard-X-rays in the energy range from 3 to 16 keV to a 100 nm scale at the SPB/SFX instrument of the European XFEL [8]. Both mirrors are elliptical cylinders, both apertures are 950 mm (sagittal) × 25 mm (tangential). These ultra-precision cylindrical optics used in synchrotron radiation systems and laser shaping devices have two characteristics. First, the shape of generatrix is not a circle but the departure from the cylindrical surface is not large. Second, the aperture is large, generally greater than 200 mm in a single direction. This type of optics is called large aperture mild acylinder in the paper. The manufacture of ultra-precision optical elements requires the support of ultra-precision surface figure measurement, but existing methods lack flexibility and are limited in aperture, which makes it impossible to realize ultra-precision measurement of large aperture acylinders. Due to the dual curvature characteristics of cylindrical mirrors, the traditional null optics in the form of rotationally symmetric spherical lenses is not applicable, and CGH is commonly used [9–13]. The diffraction structure made on the substrate of CGH can generate phase modulation for the incident test wavefront, and transform it into the wavefront matching the measured surface shape nominally. Since the equidistant surface of a cylinder is still a cylinder, a series of cylinders with different curvature radii can be tested with the same CGH, as long as the focal line of the test wavefront coincides with the axis of the measured cylinder. The ultra-precision measurement of the acylindrical surface can also use CGH to compensate the aberration
∗ Corresponding author. E-mail addresses: [email protected] (L. Kong), [email protected] (S. Chen).
https://doi.org/10.1016/j.optcom.2019.124526 Received 1 July 2019; Received in revised form 30 August 2019; Accepted 6 September 2019 Available online 9 September 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
L. Kong and S. Chen
Optics Communications 455 (2020) 124526
Fig. 1. The optical path of near-null test (a) Sagittal angle of view (b) Axial angle of view.
and realize the null interference test. However, unlike the cylindrical surface, the acylindrical CGH has no surface shape adaptability. It is designed for accurate aberration balance of a specific surface shape. The same CGH can only be used for a single acylindrical surface. This ‘‘one-to-one’’ customized CGH cannot be used for another acylindrical surface with different parameters, resulting in waste of time and money. In addition, limited by the aperture of CGH and interferometer, it is not suitable for measurement of large aperture acylindrical surfaces. Subaperture stitching [14] is a common method to extend the lateral range of measurement by dividing the large aperture into a series of smaller subapertures. The subapertures are scanned and measured sequentially, which inevitably introduces uncertain misalignment to each subaperture. Therefore the misalignment-induced aberrations of the subaperture must be decoupled from the surface error before we stitch all subaperture measurements together. In order to solve the above problems, a near-null subaperture stitching method is proposed in this paper. A general cylindrical CGH is used for near-null test of acylindrical surface, which compensates most aberrations and ensures that the interference fringes can be resolved by a standard interferometer. Ray tracing and simulation are used to correct the residual wavefront. The misalignment-induced aberration model of cylinders are revised to make it suitable for acylinder measurement. And by combining subaperture stitching technology, the large aperture acylinder surface error can be measured and reconstructed. This paper first introduces the principle of the proposed method in the second part, and then uses the proposed method to realize measurement of an elliptical cylindrical mirror with full aperture of 240 mm (sagittal) × 350 mm (tangential) in the third part. Finally, the cross test experiment verifies the validity and precision of the proposed method.
test expands the dynamic range of measurement by compensating most aberrations. It is the main method to realize generally flexible measurement of ultra-precision surfaces. The optical path of near-null test system for the acylinder based on cylindrical CGH is shown in Fig. 1. The collimated beam with a wavelength of 632.8 nm is split into reflected reference beam and transmitted test beam through transmission flat (TF) of interferometer. The test wavefront is converted to cylindrical wavefront by CGH diffraction. If the focal line of the wavefront coincides with the axis of best-fit cylinder, most normal aberrations of the measured acylinder will be compensated, and the residual wavefront aberration corresponds to the deviation of the acylinder from the best-fit cylinder. The beam reflected from the measured surface is hence slightly deviated from the original incident path (not common path), carrying shape information and residual wavefront aberration, and then interferes with the reference beam after passing through CGH and TF again. Finally, the interference is imaged on the CCD of the interferometer for data acquisition with phase shifting. Since the phase map is the result of combined effect of the residual wavefront aberration and the measured surface shape, residual wavefront aberration must be removed in order to obtain accurate surface figure. In the test system shown in Fig. 1, the residual wavefront aberration is obtained by tracing rays to the exit pupil of the interferometer. It is then removed from the test wavefront and the shape error of the measured surface can be obtained. In this method, when the test beam reflected from the measured surface passes through CGH again, the retrace error caused by non-common path is also included in the residual wavefront aberration. If the parameters of the internal optics in the interferometer are known, we can continue to trace rays to the image plane of the interferometer and include the retrace error caused by the internal optics. As shown in Fig. 2, taking the measured mirror surface as an example, the origin of section coordinates is established at the intersection of the focal line and the section of the wavefront. It is assumed that the curvature center of best-fit cylinder coincides with the origin, and the measured surface is symmetrical about the optical axis. The phase map obtained from the interferometer gives a set of discrete data. The ray corresponding to the pixel on the CCD imaging
2. Principle 2.1. Principle of near-null test for acylinders Since the CGH for the cylindrical surface has become an off-theshelf product, and the wavefront generated by the cylindrical CGH deviates mildly from the desired acylindrical wavefront, we can use the cylindrical CGH to realize near-null test of acylinders. Near-null 2
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Optics Communications 455 (2020) 124526
Fig. 2. Optical path sectional view of near-null test.
area is traced to the CGH diffractive surface. Let the points 𝑎1 ,a2 ,. . . ,a𝑛 of the CGH diffractive surface section are evenly spaced with spacing 𝑑1 , and 𝑛 is the number of pixels in the 𝑥-axis direction. Set up the projection plane behind the tested surface. 𝑏1 ,b2 ,. . . ,b𝑛 is the intersection point between the projection plane and the rays where 𝑎1 ,a2 ,. . . ,a𝑛 are located. Since the sides of similar triangles are proportional, 𝑏1 ,b2 ,. . . ,b𝑛 are also evenly spaced and the spacing is 𝑑2 . set the distance of 𝑏1 and 𝑏𝑛 as 𝐻, then 𝑑2 = 𝐻/(𝑛 − 1). If the size of the sagittal direction of the measured area is ℎ, conic constant is 𝑘, the section curve of the measured mirror can be expressed as: √
𝑧=𝑟− 1+
𝑥2 𝑟
1 − (1 + 𝑘)
( )2 𝑥 𝑟
, (−
ℎ ℎ ≤𝑥≤ ) 2 2
(1)
Fig. 3. Schematic of subaperture stitching.
If the distance from the projection plane to the origin is 𝐷, the line 𝑂𝑏𝑖 can be expressed as: 𝑧=
𝐷 𝐻 2
− (𝑖 − 1) 𝑑2
𝑥, (𝑖 = 1, 2, … , 𝑚; 𝑚 ≤
𝑛 ) 2
(2)
𝐻 can be represented by ℎ and 𝐷. According to Eqs. (1) and (2), the coordinate 𝐴𝑖 (𝑥𝑖 ,z 𝑖 ) of the intersection point between the section curve and the line 𝑂𝑏𝑖 can be obtained. When the acylindrical surface is measured with cylindrical CGH, if the maximum sparsity of the interference fringe is taken as the optimization objective, the curve radius 𝑅 of best-fit circle of the acylindrical section can be obtained in the lens design software, Then 𝑅 = r + 𝛥𝑧, where 𝛥𝑧 is the distance between the acylindrical section and the best-fit circle in the direction of the optical axis. The residual wavefront aberration can be expressed as √ ( ) 𝑂𝑃 𝐷𝑖 = 2 𝑅 − ||𝑂𝐴𝑖 || = 2(𝑟 + 𝛥𝑧 − 𝑥2𝑖 + 𝑧2𝑖 ) (3)
Fig. 4. Schematic of adjustment errors.
from two tests in the overlapping region should be consistent, but they are always different [16,17] in real test due to the misalignmentinduced aberration. The subaperture stitching algorithm is to separate the effects of errors such as misalignment, and minimize the deviation of the overlapping regions. Application of subaperture stitching test technology in plane, spherical surface and aspheric surface has been extensively studied. For example, QED Technologies announced SSI-AR series subaperture stitching interferometer [18–20] for aspheres in the last decade. However, the misalignment-induced aberration model and stitching algorithms are mostly aimed at rotationary symmetric optics, which cannot be directly applied to cylinders and acylinders. Chen and his group proposed a subaperture connection method for fringe projection profilometry for full cylinders [21,22]. In addition, Peng et al. [23,24] established a mathematical model for misalignment-induced aberration based on optical path difference analysis and Legendre polynomials, developed a stitching algorithm for cylindrical surfaces, and realized stitching interferometry for cylindrical optics with large numerical aperture [25, 26]. Recently we compared typical methods for stitching optimization
According to Eq. (3), matrix 𝑈𝑂𝑃 𝐷 , 𝑉𝑂𝑃 𝐷 and 𝑊𝑂𝑃 𝐷 are established, representing the 𝑥, 𝑦 coordinates and corresponding residual wavefront aberration respectively. After tracing rays to the exit pupil of the interferometer and separating the residual wavefront aberration from the test wavefront, the acylinder surface error of near-null test can be reconstructed. 2.2. Principle of subaperture stitching test for large aperture acylinder Due to the limitation of CGH aperture and F/number of the test wavefront, the interferometric test based on CGH cannot obtain the full aperture surface figure of the large aperture acylindrical surface through a single measurement. Subaperture stitching interferometry(SSI) reduces the cost and complexity of testing large optics [15]. The scheme of subaperture stitching is shown in Fig. 3, where subapertures 1, 2 and 3 are the areas measured by the interferometer three times on the large aperture acylinder. The shaded areas are overlapped areas. Theoretically, the wavefront phase values obtained 3
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Optics Communications 455 (2020) 124526
Fig. 5. The principle of the proposed method.
Fig. 6. 3D layout in lens design software.
and demonstrated that the configuration-based stitching algorithm can also be applied to cylinders with null optics [27]. Cylindrical mirror of 200 mm long in tangential direction was successfully measured and polished to accuracy of 0.1 μm level. For acylinders at near-null test condition, the misalignment-induced aberration model and the stitching algorithm need to be revised. Since the mild acylindrical surface is measured at near-null condition with relatively small numerical aperture, the influence of highorder misalignment aberrations can be ignored, and only low-orders need to be considered. As shown in Fig. 4, the 𝑥 axis coincides with the focal line of cylindrical test beam. The origin of the coordinate system O-xyz is set at the midpoint of the focal line. The point 𝑃 is any point on the ideal position of the measured acylindrical surface, whose coordinate is (𝑥, 𝑦, 𝑧), and the point 𝑃 ′ is the corresponding point at the actual position of the measured acylindrical surface, whose coordinates is (𝑥′ , 𝑦′ , 𝑧′ ). Different from the cylindrical surface, in addition to the four misalignments which are translation 𝑡𝑦 along the 𝑦 axis, translation 𝑡𝑧 along the 𝑧 axis, rotation 𝜃𝑦 around the 𝑦 axis and rotation 𝜃𝑧 around the 𝑧 axis, the acylindrical surface also needs to consider the rotation 𝜃𝑥 around the 𝑥 axis. Misalignment aberrations induced by 𝜃𝑦 , 𝜃𝑧 , 𝑡𝑦 and 𝑡𝑧 have been derived by Peng et al. [23], so we need only to derive the aberration induced by 𝜃𝑥 . Let the intersection point of the section and the 𝑥 axis be 𝑂𝑖 , and let the intersection point of the section and the 𝑥′ axis corresponding to the actual position of the best-fit cylinder be 𝑂𝑖′ . According to the double optical path difference theory [28,29], optical path difference (OPD) from the point 𝑃 to the point 𝑃 ′ can be represented as follows: ( ) 𝑂𝑃 𝐷 = 2 ||𝑂𝑖 𝑃 ′ || − ||𝑂𝑖 𝑃𝑖 || + ||𝑂𝑖 𝑂𝑖′ ||𝑧 (4)
Fig. 7. Interferogram (a) and residual wavefront error (b) obtained by simulation.
According to the rigid transformation principle [30], the coordinate conversion relationship between the point 𝑃 ′ and the point 𝑃 caused by misalignment 𝜃𝑥 can be expressed as follows: ⎡𝑥′ ⎤ ⎡1 ⎢𝑦′ ⎥ = ⎢0 ⎢ ′⎥ ⎢ ⎣𝑧 ⎦ ⎣0
0 cos 𝜃𝑥 − sin 𝜃𝑥
0 ⎤ ⎡𝑥⎤ sin 𝜃𝑥 ⎥ ⎢𝑦⎥ ⎥⎢ ⎥ cos 𝜃𝑥 ⎦ ⎣𝑧⎦
Fig. 8. Subaperture partition of near-null test.
The OPD caused by misalignment 𝜃𝑥 can be expressed as: (√ ) √ ( )2 ( )2 √ 𝑂𝑃 𝐷𝜃𝑥 = 2 𝑦 + 𝑧𝜃𝑥 + 𝑧 − 𝑦𝜃𝑥 − 𝑦2 + 𝑧2 = 2𝑅( 1 + 𝜃𝑥2 − 1)
(5)
Since 𝜃𝑥 is sufficiently small, cos𝜃𝑥 ≈ 1, sin𝜃𝑥 ≈ 𝜃𝑥 , Eq. (5) can be rewritten as: ⎡𝑥′ ⎤ ⎡1 ⎢𝑦′ ⎥ = ⎢0 ⎢ ′⎥ ⎢ ⎣𝑧 ⎦ ⎣0
0 cos 𝜃𝑥 − sin 𝜃𝑥
0 ⎤ ⎡𝑥⎤ ⎡ 𝑥 ⎤ sin 𝜃𝑥 ⎥ ⎢𝑦⎥ ≈ ⎢ 𝑦 + 𝑧𝜃𝑥 ⎥ ⎥⎢ ⎥ ⎢ ⎥ cos 𝜃𝑥 ⎦ ⎣𝑧⎦ ⎣−𝑦𝜃𝑥 + 𝑧⎦
(7) Combining the misalignment–induced aberration model Eq. (7) with stitching algorithm for cylindrical surface [23–27], it can be used for acylindrical surface stitching. Since the deviation of the surface shape
(6)
4
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Optics Communications 455 (2020) 124526
Fig. 9. The layout of near-null subaperture stitching system.
Fig. 10. Interferogram of the four subapertures based on near-null test.
in the subaperture from the cylindrical surface is smaller than the full aperture, a better local testing result can be obtained. At the same time, in order to avoid the error accumulation caused by subaperture stitching one by one, the global stitching strategy [31] is adopted. After minimizing the sum of squared phase deviation of all adjacent subaperture overlapping regions, full aperture stitching result of the measured acylinder is obtained. To minimize the misalignment error, alignment pattern is fabricated along with the null pattern on the CGH substrate. As collimated beam is used, the alignment pattern is simply linear grating ensuring the CGH is well aligned with the interferometer at nominal tilting angles. The CGH substrate error in thickness or refraction index variation can be calibrated out by testing the zero order wavefront error with a reference flat artifact. The manufacturing errors of CGH include pattern distortion, etching depth variations [32,33], etc. The photomask fabrication process including laser writing and ion etching can typically reduce error to a level of 10 nm.
the misalignment-induced aberrations and CGH errors, etc. After the residual wavefront aberration, CGH errors and misalignment-induced aberrations of each subaperture are separated, the full aperture surface figure can be obtained. 3. Experiment 3.1. Simulation The mirror to be tested is a concave elliptical cylindrical reflector, the conic constant 𝑘 = −0.7432, clear aperture 180 mm (sagittal) × 300 mm (tangential), and vertex curve radius 𝑅 = 1750 mm. The null pattern region of CGH for the cylindrical surface is 100 mm × 100 mm, the distance from the center of null pattern region to the focal line is 1012.3 mm. The 1st order of diffraction is used with a 5◦ tilt carrier added to isolate the disturbance orders (see Fig. 6). 3.2. Application experiment
2.3. Subaperture stitching for large aperture acylinders based on near-null test
The Zygo GPI XP/D 6-inch interferometer was used for the test. The measured subaperture can cover the effective aperture in the sagittal direction. While the clear aperture in the tangential direction is 300 mm, 4 subapertures are partitioned as shown in Fig. 8. The subaperture spacing is 67 mm, the overlap area is 33 mm, and the overlap rate is 33%. The coverage area of the subaperture stitching testing is 181.6 mm (sagittal) × 300 mm (tangential). The layout of the testing system built according to above scheme is shown in Fig. 9. The interferograms of the tested four subapertures are shown in Fig. 10, and the corresponding wavefront error in single pass is shown in Fig. 11. The subaperture wavefront errors after removing the misalignment-induced aberrations using the Zygo software MetroPro
The combination of near-null test and subaperture stitching can provide an economical and quick test for large acylindrical surface. The principle is shown in Fig. 5. For the large aperture acylindrical optics, the subaperture is partitioned according to the aperture and F/number of selected cylindrical CGH. According to the maximum residual wavefront slope and the maximum spatial frequency of interference fringe [34], it is judged whether the near-null test of subaperture can be realized by using the selected CGH. If it can be tested, the problems need to be solved include ray tracing, removal of the residual wavefront aberration, 5
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Optics Communications 455 (2020) 124526
Fig. 11. Wavefront error of the four subapertures based on near-null test.
Fig. 12. Subaperture wavefront error after removing misalignment-induced aberrations.
Fig. 13. Surface figure of the clear aperture.
are shown in Fig. 12. They are similar to the simulation results of residual wavefront error shown in Fig. 7(b). The difference is ascribed to environmental noise, surface error of the tested mirror and the CGH error, etc. Fig. 13 shows the surface figure of the clear aperture obtained by using the proposed method of subaperture stitching for large aperture acylinders based on near-null test.
Fig. 14. Subaperture partition of the null testing.
The coverage area of the subaperture stitching test is 252 mm (sagittal) × 350 mm (tangential).
3.3. Cross test
The optical path is shown in Fig. 15, and the layout of the null test system is shown in Fig. 16.
For the purpose of cross test, another customized CGH is fabricated for null test of the same acylinder. The validity and precision of the proposed method can then be verified by comparison with the null testing results. The CGH null pattern is 124 mm (sagittal) × 120 mm (tangential), covering 252 mm (sagittal) × 120 mm (tangential) area of the acylindrical surface. The center of the CGH is 850 mm away from the tested acylinder, and the frequency carrier 5◦ tilt. Similarly, the single measurable aperture cannot cover the full aperture range of the mirror under test, and the subaperture stitching test is required along the tangential direction. As shown in Fig. 14, evenly spaced 4 subapertures with spacing of 76.67 mm are arranged. The width of overlap area is 43.33 mm, and the overlap rate is 36.24%.
The wavefront error of the subapertures after removing the misalignment-induced aberrations is shown in Fig. 17. According to the stitching algorithm for acylinders, the surface figure of the clear aperture is shown in Fig. 18. By comparing Figs. 13 and 18, the surface shape error obtained by proposed near-null test is consistent with the null test result and both PV and RMS values are comparable. Furthermore, the structural similarity index [35] (SSIM Index) is introduced to evaluate the similarity of two surface shape errors. The SSIM Index is defined as follows: 𝑆𝑘 (m, n) = 6
(2𝜇𝑚 𝜇𝑛 + 𝐶1 )(2𝜎𝑚𝑛 + 𝐶2 ) 2 + 𝜇 2 + 𝐶 )(𝜎 2 + 𝜎 2 + 𝐶 ) (𝜇𝑚 1 2 𝑛 𝑚 𝑛
(8)
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Optics Communications 455 (2020) 124526
Fig. 15. Diagram of the null test system.
Fig. 16. The layout of the null test system.
where 𝜇𝑚 and 𝜇𝑛 are the average gray of the two images, 𝜎𝑚 and 𝜎𝑛 are the standard deviations, respectively. Constants 𝐶1 = (𝐾1 𝐿)2 , 𝐶2 = (𝐾2 𝐿)2 , 𝐾1 ≪ 1, 𝐾2 ≪ 1. Generally set 𝐾1 = 0.01, 𝐾2 = 0.03. 𝐿 is the image gray scale range. For 8-bit gray map 𝐿 = 255. The two surface shape errors [36] of Figs. 13 and 18 are registered to each other to eliminate the errors caused by image misalignment and scaling. Calculation shows the SSIM index is S = 0.8901. It means that the two results are quite similar in error distribution, which verifies the validity and precision of the proposed method. On the other hand, both figures show slightly visible stitching traces. The traces indicate inconsistency between the neighboring subapertures. As the subapertures have been measured with careful control of noise and then optimally stitched together by minimizing the overlapping inconsistency, we ascribe these traces probably to systematic error introduced by the CGH substrate. Although we have tried to calibrate out the substrate error through measuring the zero-order transmitted wavefront error, the difference between zero-order and first order in use results in residual systematic error and finally contributes to the subaperture traces in the stitching map. A data fusion strategy combining multiple measurements at the same pixel or filtering techniques are available to suppress the stitching traces. Compared with the conventional test method, the proposed method in the paper possesses the following major advantages. It overcomes the shortcomings of ‘‘one-to-one’’ customized CGH for acylinders, such as long production cycle, high cost, and poor versatility. At the same time, commercial cylindrical CGH is readily available, which can be used to test a series of acylinders. Combined with subaperture stitching, it becomes an economical and fast test method for large aperture optics. However, the proposed method can only test acylinders with a small deviation from the best-fit cylinder. Moreover if the stitching in the sagittal direction is required, the high-order form of misalignmentinduced aberrations of acylindrical surface should be studied, and the
model should be modified accordingly. As we know, the spatial frequency of interference fringes is proportional to the slope of wavefront. The greater slope of wavefront, the higher spatial frequency of the fringes. While the fringes must be resolvable, the dynamic range of measurement is thus basically limited by the Nyquist sampling frequency of the interferometer detector. This limit is quite similar to non-null test of mild aspheres with a spherical interferometer which has been discussed in extensively [37]. Although interferometers with higher resolution detectors are claimed to be able to resolve up to 200 fringes, the dynamic range of non-null test is usually limited to tens of waves considering the varying slope of the aspheric departure. It also applies to the near-null test proposed in this paper. On the other hand, advanced non-null test such as the sub-Nyquist interferometry [37] can be employed to extend the dynamic range. As the acylinder is aspherical in only one dimension, it is quite clever to use a yawing CGH as variable optical null which can generate larger aberrations for measuring steep acylinders [38]. However, the proposed method in this paper is mainly applied to mild acylinders by using a stationary CGH, which is easier to be implemented and integrated into the subaperture test system. 4. Conclusion In order to solve the problems of large aperture mild acylindrical surface test, this paper combines near-null test and subaperture stitching methods. It uses cylindrical CGH to achieve near-null compensation for various acylindrical surfaces with mild aspherical departure, and applies the subaperture stitching to extend the lateral range of measurement. Using the proposed method, a concave cylindrical mirror was measured and evaluated by stitching 4 near-null subapertures. Comparing with the result of subaperture stitching with customized null CGH for the same mirror, the validity and precision of the proposed method are verified. 7
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Optics Communications 455 (2020) 124526
Fig. 17. Null testing phase diagram after removing misalignment aberrations.
[14] D. Malacara, Optical Shop Testing, third ed., Jolm Wiley & Sons, Inc., New Jersey, 2007. [15] J.C. Kim, Polynomial fit of interferograms, Appl. Opt. 21 (24) (1982) 4521–4525. [16] Rongzhu Zhang, Chunlin Yang, Qiao Xu, et al., High Power Laser Part. Beams 16 (7) (2004) 789–882. [17] Mingyi Zhang, Xinnan Li, Accuracy analysis of stitching interferometry for test of large-diametermirror, J. Appl. Opt. 27 (5) (2006) 446–449. [18] P. Murphy, G. Forbes, J. Fleig, P. Dumas, M. Tricard, Stitching interferometry: A flexible solution for surface metrology, Opt. Photonics News 14 (2003) 38–43. [19] G. Forbes, P.R. Dumas, J.F. Fleg, P.E. Murphy, Subaperture stiching interferometer for versatile and accurate surface metrology, in: Optical Fabrication and Testing, 2002. [20] P. Murphy, J. Fleig, G. Forbes, D. Miladinovic, G. DeVries, S. ÓDonohue, Subaperture stiching interferometer for testing mild aspheres, in: SPIE, 2006, p. 62930J. [21] H. Guo, M. Chen, Multiview connection technique for 360-deg 3D measurement, Opt. Eng. 42 (2003) 900–905. [22] H. He, M. Chen, H. Guo, Y. Yu, Novel multiview connection method based on virtual cylinder for 3-d surface measurement, Opt. Eng. 44 (2005) 083605–1–6. [23] J. Peng, D. Ge, Y. Yu, K. Wang, M. Chen, Method of misalignment aberrations removal in null test of cylindrical surface, Appl. Opt. 52 (2013) 7311–7323. [24] J. Peng, Y. Yu, H. Xu, Compensation of high-order misalignment aberrations in cylindrical interferometry, Appl. Opt. 53 (2014) 4947–4956. [25] J. Peng, H. Xu, Y. Yu, M. Chen, Stitching interferometry for cylindrical optics with large angular aperture, Meas. Sci. Technol. 26 (2015) 025204. [26] J. Peng, Q. Wang, X. Peng, Y. Yu, Stitching interferometry of high numerical aperture cylindrical optics without using a fringe-nulling routine, J. Opt. Soc. Amer. 32 (2015) 1964–1972. [27] Shanyong Chen, Shuai Xue, Guilin Wang, Ye Tian, Subaperture stitching algorithms: A comparison, Opt. Commun. 390 (2017) 61–71. [28] D. Liu, Y. Yang, C. Tian, et al., Practical methods for retrace error correction in nonnull aspheric testing, Opt. Express 17 (9) (2009) 7025–7035. [29] Xiaokun Wang, Lihui Wang, Weijie Deng, et al., Measurement of large aspheric mirros by non-null testing, Opt. Precis. Eng. 19 (3) (2011) 520–528. [30] Chongho Wang, Coordinate turning method between motion values in systems of rigid bodies, J. Changsha Railway Inst. 7 (4) (1989) 99–108. [31] M. Otsubo, K. Okada, J. Tsujiuchi, Measurement of large plane surface by connecting small-aperture interferograms, Opt. Eng. 33 (2) (1994) 608–613. [32] P. Zhou, J.H. Burge, Fabrication error analysis and experimental demonstration for computer-generated holograms, Appl. Opt. 46 (5) (2007) 657–663. [33] Y.C. Chang, P. Zhou, J.H. Burge, Analysis of phase sensitivity for binary computer-generated holograms, Appl. Opt. 45 (18) (2006) 4223–4234. [34] Xiaochen Meng, Qun Hao, Qiudong Zhu, Yao Hu, Influence of interference fringe’s spatial frequency on the phase measurement accuracy in digital moiré phase-shifting interferometry, Chin. J. Lasers 38 (10) (2011) 1008008–1–1008008–6. [35] Z. Wang, A. Bovik, H. Sheikh, E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process. 13 (4) (2004) 600–612. [36] S. Chen, Y. Dai, X. Nie, S. Li, Parametric registration of cross test error maps for optical surfaces, Opt.Commum. 346 (2015) 158–166. [37] Andrew E. Lowman, John E. Greivenkamp, Modeling an interferometer for non-null testing of aspheres, Proc. SPIE 2536 (1995) 139–147. [38] Junzheng Peng, Dingfu Chen, Hongwei Guo, Jingang Zhong, Yingjie Yu, Variable optical null based on a yawing CGH for measuring steep acylindrical surface, Opt. Express 26 (2018) 20306–20318.
Fig. 18. Null test surface figure of the clear aperture.
Acknowledgments This research is supported by Natural Science Foundation of China (51975578) and Science Challenge Program of China (TZ2018006). References [1] P. Kirkpatrick, A.V. Baez, Formation of optical images by X-ray, J. Opt. Soc.Am 38 (9) (1948) 766–774. [2] S. Matsuyama, H. Mimura, H. Yumoto, et al., Development of scanning x-ray fluorescence microscope with spatial resolution of 30 nm using Kirkpatrick—Baez mirror optics, J.rev.Sci.Instrnm 77 (10) (2006) 14. [3] F.J. Marshall, J.A. Ortel, Framed 16-image Kirkpatrick-Baez microscope for blaser-plasma x-ray emission, Rev.Sci.Instrnm. 75 (10) (2004) 4045–4047. [4] T. Jach, A.S. Bakulin, S.M. Durbin, et al., Variable magnification with Kirkpatrick—Baez optics for synchrotron x-ray microscopy, J. Res. Natl. Inst. Stand. Techn01. 111 (3) (2006) 219–225. [5] Yosihio Suzuki, et al., X-ray microscope with a pair of elliptical mirrors, Japan. J. Appl. Phys. 28 (9) (1989) 1660–1662. [6] Jiahua Chen, et al., KB mirror system of X-ray photo-emission electron microscope beamline, Opt. Precission Eng. 25 (11) (2017) 2810–2816. [7] Kazuto Yamauchi, Makina Yabashi, Haruhiko Ohashi, Nanofocusing of X-ray freeelectron lasers by grazing-incidence reflective optics, Synchrotron Rad. 22 (2015) 592–598. [8] F. Siewert, J. Buchheim, G. Gwalt, R. Bean, A.P. Mancuso, On the characterization of a 1 m long, ultra-precise KB-focusing mirror pair for European XFEL by means of slope measuring deflectometry, Rev. Sci. Instrum. 90 (2019) 021713. [9] P. Tam, K. Gross, J. Bogan, Interferometric testing of cylinder optics using computer generated hologram (CGH), Proc. SPIE 3134 (1997) 162–166. [10] J. Schwider, N. Lindlein, R. Schreiner, J. Lamprecht, Grazing-incidence test for cylindrical microlenses with high numerical aperture, Opt. A 4 (2002) S10–S16. [11] J. Lamprecht, N. Lindlein, J. Schwider, Null test measurement of high-numericalperture cylindrical microlenses in transmitted light, Proc. SPIE 5180 (2004) 253–260. [12] K. Mantel, N. Lindlein, J. Schwider, Simultaneous characterization of the quality and orientation of cylindrical lens surfaces, Appl. Opt. 44 (2005) 2970–2977. [13] K. Mantel, J. Lamprecht, N. Lindlein, J. Schwider, Absolute calibration in grazing incidence interferometry via rotational averaging, Appl. Opt. 45 (2006) 3740–3745.
8
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Subaperture stitching test for large aperture mild acylinders Lingwei Kong, Shanyong Chen ∗ College of Artificial Intelligence, National University of Defense Technology, Changsha, Hunan, 410073, China Hunan Key Laboratory of Ultra-Precision Machining Technology, Changsha, Hunan, 410073, China Laboratory of Science and Technology on Integrated Logistics Support, National University of Defense Technology, Changsha, Hunan, 410073, China
ARTICLE
INFO
Keywords: Large aperture acylinder Subaperture stitching Near-null Surface reconstruction
ABSTRACT Large aperture acylinders are typical optical elements in synchrotron radiation systems and laser shaping devices. The manufacture of such ultra-precision optical elements requires ultra-precision measurement of the surface figure. The use of computer-generated hologram (CGH) customized for the acylindrical surface can only achieve a ‘‘one-to-one’’ test. Limited by the aperture of CGH, it also cannot meet the requirements for large aperture acylinders. In order to realize flexible test of large aperture acylinders with mild aspherical departure, a method of surface figure reconstruction for subaperture measurements under near-null test is proposed. It is based on ray tracing and rigid body transformation. By using a general CGH for cylindrical surface, near-null test for various acylindrical surfaces can be realized. Subaperture stitching test can overcome the problem of the limited CGH aperture which cannot meet the requirement of large aperture acylinders. Finally an elliptical cylindrical mirror with an aperture of 240mm (sagittal) × 350mm (tangential) is measured. And the cross test is realized by using a customized CGH to verify the validity and precision of the proposed method.
1. Introduction Mild acylinder refers to the aspherical cylinder whose generatrix is close to the circle, i.e. with mild aspherical departure. Similar to the cylindrical surface, mild acylinder has different curvatures in tangential and sagittal directions, respectively. It is often used for correcting astigmatic aberration, line focusing, and line imaging. Compared with the cylinder, acylinder introduces conic constants and even high-order terms, which indicates more degrees of freedom for optimization design and better performance can be achieved. So it is more and more widely used in synchrotron radiation systems and laser shaping devices. For example, the KB (Kirkpatrick–Baez) mirror system [1] which is composed of a pair of acylindrical mirrors has been widely used due to its advantages of correcting astigmatism, simple structure, and relatively low processing and assembly requirements [2–4]. For the first time, Suzuki et al. constructed a KB mirror system using mutually perpendicular elliptical cylindrical reflectors [5], which can better eliminate the spherical aberration compared to the KB system constructed by conventional cylindrical surfaces. The KB mirror system of PEEM in Shanghai synchrotron radiation facility [6] uses two ultra-precision elliptical cylindrical mirrors with aperture of 250 mm (sagittal) × 40 mm (tangential), 450 mm (sagittal) × 30 mm (tangential), and the surface slope error is better than 5 μrad. The optical configuration of focusing system of SPring-8 angstrom compact free-electron laser (SACLA) adopts two-stage KB mirror system [7], which uses a total of four elliptical cylindrical reflectors. Recently, the European X-ray Free
Electron Laser (XFEL) has successfully produced its first X-ray photon pulse trains. A KB mirror pair is designed to focus hard-X-rays in the energy range from 3 to 16 keV to a 100 nm scale at the SPB/SFX instrument of the European XFEL [8]. Both mirrors are elliptical cylinders, both apertures are 950 mm (sagittal) × 25 mm (tangential). These ultra-precision cylindrical optics used in synchrotron radiation systems and laser shaping devices have two characteristics. First, the shape of generatrix is not a circle but the departure from the cylindrical surface is not large. Second, the aperture is large, generally greater than 200 mm in a single direction. This type of optics is called large aperture mild acylinder in the paper. The manufacture of ultra-precision optical elements requires the support of ultra-precision surface figure measurement, but existing methods lack flexibility and are limited in aperture, which makes it impossible to realize ultra-precision measurement of large aperture acylinders. Due to the dual curvature characteristics of cylindrical mirrors, the traditional null optics in the form of rotationally symmetric spherical lenses is not applicable, and CGH is commonly used [9–13]. The diffraction structure made on the substrate of CGH can generate phase modulation for the incident test wavefront, and transform it into the wavefront matching the measured surface shape nominally. Since the equidistant surface of a cylinder is still a cylinder, a series of cylinders with different curvature radii can be tested with the same CGH, as long as the focal line of the test wavefront coincides with the axis of the measured cylinder. The ultra-precision measurement of the acylindrical surface can also use CGH to compensate the aberration
∗ Corresponding author. E-mail addresses: [email protected] (L. Kong), [email protected] (S. Chen).
https://doi.org/10.1016/j.optcom.2019.124526 Received 1 July 2019; Received in revised form 30 August 2019; Accepted 6 September 2019 Available online 9 September 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
L. Kong and S. Chen
Optics Communications 455 (2020) 124526
Fig. 1. The optical path of near-null test (a) Sagittal angle of view (b) Axial angle of view.
and realize the null interference test. However, unlike the cylindrical surface, the acylindrical CGH has no surface shape adaptability. It is designed for accurate aberration balance of a specific surface shape. The same CGH can only be used for a single acylindrical surface. This ‘‘one-to-one’’ customized CGH cannot be used for another acylindrical surface with different parameters, resulting in waste of time and money. In addition, limited by the aperture of CGH and interferometer, it is not suitable for measurement of large aperture acylindrical surfaces. Subaperture stitching [14] is a common method to extend the lateral range of measurement by dividing the large aperture into a series of smaller subapertures. The subapertures are scanned and measured sequentially, which inevitably introduces uncertain misalignment to each subaperture. Therefore the misalignment-induced aberrations of the subaperture must be decoupled from the surface error before we stitch all subaperture measurements together. In order to solve the above problems, a near-null subaperture stitching method is proposed in this paper. A general cylindrical CGH is used for near-null test of acylindrical surface, which compensates most aberrations and ensures that the interference fringes can be resolved by a standard interferometer. Ray tracing and simulation are used to correct the residual wavefront. The misalignment-induced aberration model of cylinders are revised to make it suitable for acylinder measurement. And by combining subaperture stitching technology, the large aperture acylinder surface error can be measured and reconstructed. This paper first introduces the principle of the proposed method in the second part, and then uses the proposed method to realize measurement of an elliptical cylindrical mirror with full aperture of 240 mm (sagittal) × 350 mm (tangential) in the third part. Finally, the cross test experiment verifies the validity and precision of the proposed method.
test expands the dynamic range of measurement by compensating most aberrations. It is the main method to realize generally flexible measurement of ultra-precision surfaces. The optical path of near-null test system for the acylinder based on cylindrical CGH is shown in Fig. 1. The collimated beam with a wavelength of 632.8 nm is split into reflected reference beam and transmitted test beam through transmission flat (TF) of interferometer. The test wavefront is converted to cylindrical wavefront by CGH diffraction. If the focal line of the wavefront coincides with the axis of best-fit cylinder, most normal aberrations of the measured acylinder will be compensated, and the residual wavefront aberration corresponds to the deviation of the acylinder from the best-fit cylinder. The beam reflected from the measured surface is hence slightly deviated from the original incident path (not common path), carrying shape information and residual wavefront aberration, and then interferes with the reference beam after passing through CGH and TF again. Finally, the interference is imaged on the CCD of the interferometer for data acquisition with phase shifting. Since the phase map is the result of combined effect of the residual wavefront aberration and the measured surface shape, residual wavefront aberration must be removed in order to obtain accurate surface figure. In the test system shown in Fig. 1, the residual wavefront aberration is obtained by tracing rays to the exit pupil of the interferometer. It is then removed from the test wavefront and the shape error of the measured surface can be obtained. In this method, when the test beam reflected from the measured surface passes through CGH again, the retrace error caused by non-common path is also included in the residual wavefront aberration. If the parameters of the internal optics in the interferometer are known, we can continue to trace rays to the image plane of the interferometer and include the retrace error caused by the internal optics. As shown in Fig. 2, taking the measured mirror surface as an example, the origin of section coordinates is established at the intersection of the focal line and the section of the wavefront. It is assumed that the curvature center of best-fit cylinder coincides with the origin, and the measured surface is symmetrical about the optical axis. The phase map obtained from the interferometer gives a set of discrete data. The ray corresponding to the pixel on the CCD imaging
2. Principle 2.1. Principle of near-null test for acylinders Since the CGH for the cylindrical surface has become an off-theshelf product, and the wavefront generated by the cylindrical CGH deviates mildly from the desired acylindrical wavefront, we can use the cylindrical CGH to realize near-null test of acylinders. Near-null 2
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Optics Communications 455 (2020) 124526
Fig. 2. Optical path sectional view of near-null test.
area is traced to the CGH diffractive surface. Let the points 𝑎1 ,a2 ,. . . ,a𝑛 of the CGH diffractive surface section are evenly spaced with spacing 𝑑1 , and 𝑛 is the number of pixels in the 𝑥-axis direction. Set up the projection plane behind the tested surface. 𝑏1 ,b2 ,. . . ,b𝑛 is the intersection point between the projection plane and the rays where 𝑎1 ,a2 ,. . . ,a𝑛 are located. Since the sides of similar triangles are proportional, 𝑏1 ,b2 ,. . . ,b𝑛 are also evenly spaced and the spacing is 𝑑2 . set the distance of 𝑏1 and 𝑏𝑛 as 𝐻, then 𝑑2 = 𝐻/(𝑛 − 1). If the size of the sagittal direction of the measured area is ℎ, conic constant is 𝑘, the section curve of the measured mirror can be expressed as: √
𝑧=𝑟− 1+
𝑥2 𝑟
1 − (1 + 𝑘)
( )2 𝑥 𝑟
, (−
ℎ ℎ ≤𝑥≤ ) 2 2
(1)
Fig. 3. Schematic of subaperture stitching.
If the distance from the projection plane to the origin is 𝐷, the line 𝑂𝑏𝑖 can be expressed as: 𝑧=
𝐷 𝐻 2
− (𝑖 − 1) 𝑑2
𝑥, (𝑖 = 1, 2, … , 𝑚; 𝑚 ≤
𝑛 ) 2
(2)
𝐻 can be represented by ℎ and 𝐷. According to Eqs. (1) and (2), the coordinate 𝐴𝑖 (𝑥𝑖 ,z 𝑖 ) of the intersection point between the section curve and the line 𝑂𝑏𝑖 can be obtained. When the acylindrical surface is measured with cylindrical CGH, if the maximum sparsity of the interference fringe is taken as the optimization objective, the curve radius 𝑅 of best-fit circle of the acylindrical section can be obtained in the lens design software, Then 𝑅 = r + 𝛥𝑧, where 𝛥𝑧 is the distance between the acylindrical section and the best-fit circle in the direction of the optical axis. The residual wavefront aberration can be expressed as √ ( ) 𝑂𝑃 𝐷𝑖 = 2 𝑅 − ||𝑂𝐴𝑖 || = 2(𝑟 + 𝛥𝑧 − 𝑥2𝑖 + 𝑧2𝑖 ) (3)
Fig. 4. Schematic of adjustment errors.
from two tests in the overlapping region should be consistent, but they are always different [16,17] in real test due to the misalignmentinduced aberration. The subaperture stitching algorithm is to separate the effects of errors such as misalignment, and minimize the deviation of the overlapping regions. Application of subaperture stitching test technology in plane, spherical surface and aspheric surface has been extensively studied. For example, QED Technologies announced SSI-AR series subaperture stitching interferometer [18–20] for aspheres in the last decade. However, the misalignment-induced aberration model and stitching algorithms are mostly aimed at rotationary symmetric optics, which cannot be directly applied to cylinders and acylinders. Chen and his group proposed a subaperture connection method for fringe projection profilometry for full cylinders [21,22]. In addition, Peng et al. [23,24] established a mathematical model for misalignment-induced aberration based on optical path difference analysis and Legendre polynomials, developed a stitching algorithm for cylindrical surfaces, and realized stitching interferometry for cylindrical optics with large numerical aperture [25, 26]. Recently we compared typical methods for stitching optimization
According to Eq. (3), matrix 𝑈𝑂𝑃 𝐷 , 𝑉𝑂𝑃 𝐷 and 𝑊𝑂𝑃 𝐷 are established, representing the 𝑥, 𝑦 coordinates and corresponding residual wavefront aberration respectively. After tracing rays to the exit pupil of the interferometer and separating the residual wavefront aberration from the test wavefront, the acylinder surface error of near-null test can be reconstructed. 2.2. Principle of subaperture stitching test for large aperture acylinder Due to the limitation of CGH aperture and F/number of the test wavefront, the interferometric test based on CGH cannot obtain the full aperture surface figure of the large aperture acylindrical surface through a single measurement. Subaperture stitching interferometry(SSI) reduces the cost and complexity of testing large optics [15]. The scheme of subaperture stitching is shown in Fig. 3, where subapertures 1, 2 and 3 are the areas measured by the interferometer three times on the large aperture acylinder. The shaded areas are overlapped areas. Theoretically, the wavefront phase values obtained 3
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Optics Communications 455 (2020) 124526
Fig. 5. The principle of the proposed method.
Fig. 6. 3D layout in lens design software.
and demonstrated that the configuration-based stitching algorithm can also be applied to cylinders with null optics [27]. Cylindrical mirror of 200 mm long in tangential direction was successfully measured and polished to accuracy of 0.1 μm level. For acylinders at near-null test condition, the misalignment-induced aberration model and the stitching algorithm need to be revised. Since the mild acylindrical surface is measured at near-null condition with relatively small numerical aperture, the influence of highorder misalignment aberrations can be ignored, and only low-orders need to be considered. As shown in Fig. 4, the 𝑥 axis coincides with the focal line of cylindrical test beam. The origin of the coordinate system O-xyz is set at the midpoint of the focal line. The point 𝑃 is any point on the ideal position of the measured acylindrical surface, whose coordinate is (𝑥, 𝑦, 𝑧), and the point 𝑃 ′ is the corresponding point at the actual position of the measured acylindrical surface, whose coordinates is (𝑥′ , 𝑦′ , 𝑧′ ). Different from the cylindrical surface, in addition to the four misalignments which are translation 𝑡𝑦 along the 𝑦 axis, translation 𝑡𝑧 along the 𝑧 axis, rotation 𝜃𝑦 around the 𝑦 axis and rotation 𝜃𝑧 around the 𝑧 axis, the acylindrical surface also needs to consider the rotation 𝜃𝑥 around the 𝑥 axis. Misalignment aberrations induced by 𝜃𝑦 , 𝜃𝑧 , 𝑡𝑦 and 𝑡𝑧 have been derived by Peng et al. [23], so we need only to derive the aberration induced by 𝜃𝑥 . Let the intersection point of the section and the 𝑥 axis be 𝑂𝑖 , and let the intersection point of the section and the 𝑥′ axis corresponding to the actual position of the best-fit cylinder be 𝑂𝑖′ . According to the double optical path difference theory [28,29], optical path difference (OPD) from the point 𝑃 to the point 𝑃 ′ can be represented as follows: ( ) 𝑂𝑃 𝐷 = 2 ||𝑂𝑖 𝑃 ′ || − ||𝑂𝑖 𝑃𝑖 || + ||𝑂𝑖 𝑂𝑖′ ||𝑧 (4)
Fig. 7. Interferogram (a) and residual wavefront error (b) obtained by simulation.
According to the rigid transformation principle [30], the coordinate conversion relationship between the point 𝑃 ′ and the point 𝑃 caused by misalignment 𝜃𝑥 can be expressed as follows: ⎡𝑥′ ⎤ ⎡1 ⎢𝑦′ ⎥ = ⎢0 ⎢ ′⎥ ⎢ ⎣𝑧 ⎦ ⎣0
0 cos 𝜃𝑥 − sin 𝜃𝑥
0 ⎤ ⎡𝑥⎤ sin 𝜃𝑥 ⎥ ⎢𝑦⎥ ⎥⎢ ⎥ cos 𝜃𝑥 ⎦ ⎣𝑧⎦
Fig. 8. Subaperture partition of near-null test.
The OPD caused by misalignment 𝜃𝑥 can be expressed as: (√ ) √ ( )2 ( )2 √ 𝑂𝑃 𝐷𝜃𝑥 = 2 𝑦 + 𝑧𝜃𝑥 + 𝑧 − 𝑦𝜃𝑥 − 𝑦2 + 𝑧2 = 2𝑅( 1 + 𝜃𝑥2 − 1)
(5)
Since 𝜃𝑥 is sufficiently small, cos𝜃𝑥 ≈ 1, sin𝜃𝑥 ≈ 𝜃𝑥 , Eq. (5) can be rewritten as: ⎡𝑥′ ⎤ ⎡1 ⎢𝑦′ ⎥ = ⎢0 ⎢ ′⎥ ⎢ ⎣𝑧 ⎦ ⎣0
0 cos 𝜃𝑥 − sin 𝜃𝑥
0 ⎤ ⎡𝑥⎤ ⎡ 𝑥 ⎤ sin 𝜃𝑥 ⎥ ⎢𝑦⎥ ≈ ⎢ 𝑦 + 𝑧𝜃𝑥 ⎥ ⎥⎢ ⎥ ⎢ ⎥ cos 𝜃𝑥 ⎦ ⎣𝑧⎦ ⎣−𝑦𝜃𝑥 + 𝑧⎦
(7) Combining the misalignment–induced aberration model Eq. (7) with stitching algorithm for cylindrical surface [23–27], it can be used for acylindrical surface stitching. Since the deviation of the surface shape
(6)
4
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Optics Communications 455 (2020) 124526
Fig. 9. The layout of near-null subaperture stitching system.
Fig. 10. Interferogram of the four subapertures based on near-null test.
in the subaperture from the cylindrical surface is smaller than the full aperture, a better local testing result can be obtained. At the same time, in order to avoid the error accumulation caused by subaperture stitching one by one, the global stitching strategy [31] is adopted. After minimizing the sum of squared phase deviation of all adjacent subaperture overlapping regions, full aperture stitching result of the measured acylinder is obtained. To minimize the misalignment error, alignment pattern is fabricated along with the null pattern on the CGH substrate. As collimated beam is used, the alignment pattern is simply linear grating ensuring the CGH is well aligned with the interferometer at nominal tilting angles. The CGH substrate error in thickness or refraction index variation can be calibrated out by testing the zero order wavefront error with a reference flat artifact. The manufacturing errors of CGH include pattern distortion, etching depth variations [32,33], etc. The photomask fabrication process including laser writing and ion etching can typically reduce error to a level of 10 nm.
the misalignment-induced aberrations and CGH errors, etc. After the residual wavefront aberration, CGH errors and misalignment-induced aberrations of each subaperture are separated, the full aperture surface figure can be obtained. 3. Experiment 3.1. Simulation The mirror to be tested is a concave elliptical cylindrical reflector, the conic constant 𝑘 = −0.7432, clear aperture 180 mm (sagittal) × 300 mm (tangential), and vertex curve radius 𝑅 = 1750 mm. The null pattern region of CGH for the cylindrical surface is 100 mm × 100 mm, the distance from the center of null pattern region to the focal line is 1012.3 mm. The 1st order of diffraction is used with a 5◦ tilt carrier added to isolate the disturbance orders (see Fig. 6). 3.2. Application experiment
2.3. Subaperture stitching for large aperture acylinders based on near-null test
The Zygo GPI XP/D 6-inch interferometer was used for the test. The measured subaperture can cover the effective aperture in the sagittal direction. While the clear aperture in the tangential direction is 300 mm, 4 subapertures are partitioned as shown in Fig. 8. The subaperture spacing is 67 mm, the overlap area is 33 mm, and the overlap rate is 33%. The coverage area of the subaperture stitching testing is 181.6 mm (sagittal) × 300 mm (tangential). The layout of the testing system built according to above scheme is shown in Fig. 9. The interferograms of the tested four subapertures are shown in Fig. 10, and the corresponding wavefront error in single pass is shown in Fig. 11. The subaperture wavefront errors after removing the misalignment-induced aberrations using the Zygo software MetroPro
The combination of near-null test and subaperture stitching can provide an economical and quick test for large acylindrical surface. The principle is shown in Fig. 5. For the large aperture acylindrical optics, the subaperture is partitioned according to the aperture and F/number of selected cylindrical CGH. According to the maximum residual wavefront slope and the maximum spatial frequency of interference fringe [34], it is judged whether the near-null test of subaperture can be realized by using the selected CGH. If it can be tested, the problems need to be solved include ray tracing, removal of the residual wavefront aberration, 5
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Optics Communications 455 (2020) 124526
Fig. 11. Wavefront error of the four subapertures based on near-null test.
Fig. 12. Subaperture wavefront error after removing misalignment-induced aberrations.
Fig. 13. Surface figure of the clear aperture.
are shown in Fig. 12. They are similar to the simulation results of residual wavefront error shown in Fig. 7(b). The difference is ascribed to environmental noise, surface error of the tested mirror and the CGH error, etc. Fig. 13 shows the surface figure of the clear aperture obtained by using the proposed method of subaperture stitching for large aperture acylinders based on near-null test.
Fig. 14. Subaperture partition of the null testing.
The coverage area of the subaperture stitching test is 252 mm (sagittal) × 350 mm (tangential).
3.3. Cross test
The optical path is shown in Fig. 15, and the layout of the null test system is shown in Fig. 16.
For the purpose of cross test, another customized CGH is fabricated for null test of the same acylinder. The validity and precision of the proposed method can then be verified by comparison with the null testing results. The CGH null pattern is 124 mm (sagittal) × 120 mm (tangential), covering 252 mm (sagittal) × 120 mm (tangential) area of the acylindrical surface. The center of the CGH is 850 mm away from the tested acylinder, and the frequency carrier 5◦ tilt. Similarly, the single measurable aperture cannot cover the full aperture range of the mirror under test, and the subaperture stitching test is required along the tangential direction. As shown in Fig. 14, evenly spaced 4 subapertures with spacing of 76.67 mm are arranged. The width of overlap area is 43.33 mm, and the overlap rate is 36.24%.
The wavefront error of the subapertures after removing the misalignment-induced aberrations is shown in Fig. 17. According to the stitching algorithm for acylinders, the surface figure of the clear aperture is shown in Fig. 18. By comparing Figs. 13 and 18, the surface shape error obtained by proposed near-null test is consistent with the null test result and both PV and RMS values are comparable. Furthermore, the structural similarity index [35] (SSIM Index) is introduced to evaluate the similarity of two surface shape errors. The SSIM Index is defined as follows: 𝑆𝑘 (m, n) = 6
(2𝜇𝑚 𝜇𝑛 + 𝐶1 )(2𝜎𝑚𝑛 + 𝐶2 ) 2 + 𝜇 2 + 𝐶 )(𝜎 2 + 𝜎 2 + 𝐶 ) (𝜇𝑚 1 2 𝑛 𝑚 𝑛
(8)
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Optics Communications 455 (2020) 124526
Fig. 15. Diagram of the null test system.
Fig. 16. The layout of the null test system.
where 𝜇𝑚 and 𝜇𝑛 are the average gray of the two images, 𝜎𝑚 and 𝜎𝑛 are the standard deviations, respectively. Constants 𝐶1 = (𝐾1 𝐿)2 , 𝐶2 = (𝐾2 𝐿)2 , 𝐾1 ≪ 1, 𝐾2 ≪ 1. Generally set 𝐾1 = 0.01, 𝐾2 = 0.03. 𝐿 is the image gray scale range. For 8-bit gray map 𝐿 = 255. The two surface shape errors [36] of Figs. 13 and 18 are registered to each other to eliminate the errors caused by image misalignment and scaling. Calculation shows the SSIM index is S = 0.8901. It means that the two results are quite similar in error distribution, which verifies the validity and precision of the proposed method. On the other hand, both figures show slightly visible stitching traces. The traces indicate inconsistency between the neighboring subapertures. As the subapertures have been measured with careful control of noise and then optimally stitched together by minimizing the overlapping inconsistency, we ascribe these traces probably to systematic error introduced by the CGH substrate. Although we have tried to calibrate out the substrate error through measuring the zero-order transmitted wavefront error, the difference between zero-order and first order in use results in residual systematic error and finally contributes to the subaperture traces in the stitching map. A data fusion strategy combining multiple measurements at the same pixel or filtering techniques are available to suppress the stitching traces. Compared with the conventional test method, the proposed method in the paper possesses the following major advantages. It overcomes the shortcomings of ‘‘one-to-one’’ customized CGH for acylinders, such as long production cycle, high cost, and poor versatility. At the same time, commercial cylindrical CGH is readily available, which can be used to test a series of acylinders. Combined with subaperture stitching, it becomes an economical and fast test method for large aperture optics. However, the proposed method can only test acylinders with a small deviation from the best-fit cylinder. Moreover if the stitching in the sagittal direction is required, the high-order form of misalignmentinduced aberrations of acylindrical surface should be studied, and the
model should be modified accordingly. As we know, the spatial frequency of interference fringes is proportional to the slope of wavefront. The greater slope of wavefront, the higher spatial frequency of the fringes. While the fringes must be resolvable, the dynamic range of measurement is thus basically limited by the Nyquist sampling frequency of the interferometer detector. This limit is quite similar to non-null test of mild aspheres with a spherical interferometer which has been discussed in extensively [37]. Although interferometers with higher resolution detectors are claimed to be able to resolve up to 200 fringes, the dynamic range of non-null test is usually limited to tens of waves considering the varying slope of the aspheric departure. It also applies to the near-null test proposed in this paper. On the other hand, advanced non-null test such as the sub-Nyquist interferometry [37] can be employed to extend the dynamic range. As the acylinder is aspherical in only one dimension, it is quite clever to use a yawing CGH as variable optical null which can generate larger aberrations for measuring steep acylinders [38]. However, the proposed method in this paper is mainly applied to mild acylinders by using a stationary CGH, which is easier to be implemented and integrated into the subaperture test system. 4. Conclusion In order to solve the problems of large aperture mild acylindrical surface test, this paper combines near-null test and subaperture stitching methods. It uses cylindrical CGH to achieve near-null compensation for various acylindrical surfaces with mild aspherical departure, and applies the subaperture stitching to extend the lateral range of measurement. Using the proposed method, a concave cylindrical mirror was measured and evaluated by stitching 4 near-null subapertures. Comparing with the result of subaperture stitching with customized null CGH for the same mirror, the validity and precision of the proposed method are verified. 7
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Fig. 17. Null testing phase diagram after removing misalignment aberrations.
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Fig. 18. Null test surface figure of the clear aperture.
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