Multiposition Rotation Interference Absolute Measurement Method for High-Precision Optical Component Surfaces

Modern optical engineering requires increasingly sophisticated interferometry methods capable of conducting subnanometer scale measurements of the large aperture, high-precision optical component surfaces. However, the accuracy of interferometry measurement is limited to the accuracy with which the surface of the reference mirror employed in the interferometer system is known, and the influence of gravity-induced deformation cannot be ignored.(is is addressed in the present work by proposing a three-flat testing method based on multiposition rotation interference absolute surface measurement technology that combines the basic theory of N-position rotation with the separability of surface wavefront functions into sums of even and odd functions. (ese functions provide the rotational symmetric components of the wavefront, which then enables the absolute surface to be reconstructed based on the N-position rotation measurements. In addition, we propose a mechanical clamping combined with computational method to compensate for the gravity-induced deformations of the flats in the multiposition rotation absolute measurements. (e high precision of the proposed absolute surface measurement method is demonstrated via simulations. (e results of laboratory experiments indicate that the combination compensation method provides the high-precision surface reconstruction outcomes. (e present work provides an important contribution for supporting the interferometry measurement of large aperture, high-precision optical component surfaces.


Introduction
e development of optical engineering technology has led to systems with increasing complexity, and the optical components are subject to increasingly stringent precision requirements. For example, the surface accuracy of largescale astronomical telescopes and lithographic lenses has reached the subnanometer scale [1]. is has generated the need for increasingly sophisticated measurement methods, such as wavelength-tuned phase-shift interferometry, that are capable of measuring the subnanometer scale surface accuracy of optical components. However, the surface accuracy of optical components is generally determined as the measured difference between the component surface and a flat reference mirror employed in the interferometer system. erefore, the accuracy of interferometry measurements is limited by the surface accuracy of the reference mirror, and the impact of reference mirror surface errors on the measurement results is not negligible when the surface accuracy of the reference mirror resides at a comparable level to that of the optical component [2,3]. Accordingly, the influence of gravity-induced deformation of the reference mirror cannot be ignored in subnanometer scale measurements. Moreover, the increasing aperture sizes of optical components require larger aperture sizes for the interferometer as well, and this increases the extent to which the reference mirrors are subject to gravity-induced deformation. e primary method employed for determining the effects of reference mirror surface irregularities on the interferometry results obtained for optical components adopts absolute measurement technology to directly measure the surface errors of the reference mirror, which can then be applied to eliminate the influence of these errors on the measurement results [4,5]. Schulz and Schwider [6] first proposed the three-flat test method in 1976, which avoids the use of a reference flat by comparing the Fizeau interferometry measurements obtained over four or more measurements for all possible two-flat permutations derived from a set of three flats. Subsequently, Fritz [7] proposed a programmable three-flat test method that represented the obtained interferogram mathematically by fitting them to rotational invariant Zernike polynomials, which were then decomposed into orthogonal basis functions, and the absolute optical surface was reconstructed mathematically by fitting lines to the basis functions using the least squares method. e approach was demonstrated to obtain a closedform solution if one of the measurements conducted included the azimuthal rotation of one of the flats by some angle. is approach was formalized by Evans and Kestner [8], who proposed the basic N-position rotation theory, which was further developed by Küchel [9] and Lin et al. [10]. In this theory, the measured wavefront is decomposed into a rotationally symmetrical component and a rotationally nonsymmetrical component, and the nonrotational symmetrical component is eliminated by azimuthal rotation through N positions to obtain the absolute optical surface information. Shi et al. [11] applied simulations and experimental analyses to compare the absolute surface reconstruction obtained using the N-position rotation absolute measurement method and an oblique incidence absolute measurement method with three different surface reconstruction algorithms. However, questions remain regarding the optimal number of azimuthal rotations N that should be employed in the method. While a greater number of rotations increase the accuracy of the approach, it also greatly increases the computational burden. Moreover, the calculation of the rotationally symmetric components is subject to error, which negatively detracts from the reconstruction accuracy. In a parallel development to N-position rotation methods, Ai and Wyant [12] proposed a surface reconstruction method similarly based on the three-flat test method with mathematically represented interferogram. However, the method employed the separability of surface wavefront functions into sums of even-odd, odd-even, eveneven, and odd-odd functions and obtained these function components by conducting interferometry measurements under eight orientations measuring three flats, where one of the flats in each measurement was rotated by 180°, 90°, and 45°with respect to the stationary flat. e even-odd and oddeven functions were first obtained, from which the eveneven function was calculated. en, the components of the odd-odd function were estimated based on the array of experimental results, and the absolute surface was reconstructed based on all available function components.
Different methods for providing highly accurate surface characterizations of optical components have been developed by directly analyzing the gravity-induced deformations of the components. For example, Wang et al. [13] extended this basic approach by physically compensating for gravityinduced deformation through the direct application of a force to the edge of a frame holding the optical component as a means of inducing a counteracting gravity-induced deformation. Here, the position and magnitude of the force were optimized based on elastic shell theory models of large aperture optical components to get the best deformation compensation effect. In addition, Dewitt et al. [14] developed equations for describing the gravity-induced deformation of optical lenses, and the effects of lens size, material, and mounting method on the deformation were analyzed. ese methods could be conceivably employed to compensate for the gravity-induced deformation of large aperture reference mirrors employed in interferometer systems used for measuring the surface accuracy of high-precision optical components. e present work addresses the limitations of past work by proposing a modified three-flat testing method based on multiposition rotation interference absolute measurement technology, which combines the N-position rotation theory with the separability of functions. e Zernike polynomials fitting algorithm was used usually. However, here, the measured wavefronts are decomposed into even-odd, oddeven, even-even, and odd-odd functions. ese functions are then employed to obtain the rotational symmetric components of the wavefronts in conjunction with an N-averaging algorithm. Finally, the absolute surface is reconstructed based on the N-position rotation measurements. In addition, we propose a mechanical clamping method to compensate for the gravity-induced deformations of the three flats in the multiposition rotation absolute measurements when the materials and dimensions of the flats are determined to be known. Different from the listed references, as well as a computational compensation method that is applicable even when the materials and/or dimensions are changed, the results of simulations demonstrate that average root-meansquare (RMS) differences between mathematically generated irregular surfaces and the surfaces reconstructed by the proposed multiposition rotation absolute measurement method are less than 0.003 nm, while the average peak-tovalley (PV) differences are less than 0.01 nm. e results of laboratory experiments indicate that the compensation method provides the high-precision surface reconstruction outcomes.

Theoretical Basis of Multiposition Rotation
Absolute Measurements e measurement principle of the N-position rotation method is illustrated in Figure 1, which involves three circular optical flats, A, B, and C, in different Fizeau interferometer configurations. We note that flat A is always on the top and flat B is always on the bottom, while flat C is on both the top (where it is referred to as C s ) and bottom positions. We also note that the flat in the top position always has a (−x, y) orientation, while the flat in the bottom position generally has a (x, y) orientation, but may have (−x, −y) or an orientation (r) that varies in uniform angular displacements Δθ; through a complete 360°rotation relative to the top flat, the rotation axis z is perpendicular to the measured surface.
For a wavefront function W(r, θ) in polar coordinates, the rotationally symmetric component W r (r, θ) can be expressed as 2 International Journal of Optics Here, W r (r, θ) is an odd-odd function. Moreover, it is rotationally invariant and is therefore constant with respect to Δθ. e integral in equation (1) can be approximated as a sequence of N discrete steps as follows: Here, the approximation introduces the uniform angular increment Δθ � 2π/N. Accordingly, the rotationally symmetric components can be obtained by rotating the wavefront function W(r, θ) by N equally spaced angular intervals Δθ. ese wavefront functions can be correlated with interferometry measurements for the three optical flat configurations illustrated in Figure 1 to obtain the following rotationally symmetric components: As discussed, B (r) represents the rotation of flat B about flat A through N equally spaced rotations of angular interval Δθ � 2π/N. Accordingly, W 5 can be expressed as follows: Here, B r (x, y) represents the rotational invariant component of the wavefront function B (x, y) of flat B. Accordingly, the odd-odd function of the flat B can be eliminated from W 5 (x, y). en, W 5 (x, y) includes only the odd-odd function of flat C (i.e., W 5oo ≈ − C oo ).
Using flip and mutual operations along the x and y axes, together with the odd and even function method, enables the odd-even, even-odd, and odd-odd function terms of the three flats to be measured. In addition, all function terms of flat C can be solved because the W 5 (x, y) wavefront contains only the odd-odd function term of flat C. Accordingly, the function term components of flats A and B can be obtained by, respectively, substituting the calculated function components of flat C into the other wavefronts. e absolute surfaces of the three flats are then obtained as follows:  International Journal of Optics 3

Simulation of Multiposition Rotation
Absolute Measurements e simulations employed three flats A (x, y), B (x, y), and C (x, y) generated by Zernike polynomials with the coefficients being set to obtain surfaces with substantially different irregularities, but with average deviations from a perfect plane which were nearly equivalent. is yielded average PV errors of 14.39 nm, 15.61 nm, and 13.88 nm for flats A, B, and C, respectively, and average RMS errors of 2.19 nm, 2.86 nm, and 2.57 nm, respectively. e surface topographies are shown in Figure 2. e proposed absolute measurement process was evaluated for the three flats based on the simulated wavefront obtained under the proposed processing.
We first evaluated the effect of the selected value of N on the simulated measurement results for values of N in the range from 3 to 12. According to the described method, the wavefront calculated by equation (4) is flipped along the x and y axes, the required components are calculated, and the absolute surfaces are obtained by equation (5). e RMS differences between the generated surface and the reconstructed surface (i.e., RMS errors) were calculated, and the distributions of the RMS errors obtained for flat A with N values of 3, 6, 9, and 12 are presented in Figures 3(a)-3(d), respectively. We note that the error distributions vary widely for N values of 3 and 6, but are reasonably stable for N values of 9 and 12, indicating that little benefit is gained by employing N � 12. e distributions of RMS errors obtained for flats A, B, and C with N � 9 are presented in Figures 4(a)-4(c), respectively. In addition, the average RMS errors were all less than 0.001 nm, while the average PV errors were all less than 0.01 nm.
We also evaluated the extent to which the average magnitude of deviations of the generated flats impacted the accuracy of the reconstructed surfaces obtained with N � 9.
is was evaluated by generating the same characteristic flats A, B, and C shown in Figure 2 using Zernike polynomials with the coefficients being set to obtain surfaces with average deviations, every group flat was nearly in the approximate interval and varied in terms of the average PV errors in the range of 10-200 nm. e reconstructed surfaces were obtained according to the proposed method, and the PV and RMS differences between the initialed and reconstructed surfaces were calculated.
e PV value varies between 0.009 nm and 0.011 nm, and the RMS value varies between 0.002 nm and 0.004 nm.
ese results are presented in Figure 5 with respect to the average deviation of the generated surfaces from the initial plane. e results demonstrate that the magnitude of the average deviation of the generated surfaces from the initial plane has a negligible effect on the accuracy of the surfaces reconstructed using the proposed method, even when the initial plane deviation is as large as 200 nm. Accordingly, the simulation results indicate that the multiposition rotation absolute surface measurement method has better accuracy under different initial surface shape error conditions.

Gravity-Induced Deformation Compensation
Model. e measured planes are placed horizontally and vertically usually, according to two types of common structures in the field of high-precision interferometry of plane shape. e influence of gravity and clamping deformation in horizontal structures is very obvious. Especially the gravitational effect plays an important role and is evenly distributed and controllable. In opposition, the gravity deformation of vertical structures has little influence and it is also difficult to accurately control the clamping deformation. In addition, it is easier to adapt a horizontal structure to the actual processing state and it is more convenient to transfer and adjust it in the process, so the horizontal structures present some advantages. e goal of gravity-induced deformation compensation is to eliminate the deformation error caused by gravity from the absolute measurement results. e deformation conditions of the top and bottom flats can be analyzed according to Figure 6. As shown in Figure 6(a), flat C exhibits deformation when it is on the top. However, Figure 6(b) indicates that flat C is nearly free of deformation when on the bottom. e asymmetrical deformations associated with these two conditions represent varying optical path length differences that introduce deviations from the ideal wavefront expressions in equation (3). However, if the materials and dimensions of the two flats are equivalent, the mechanical clamping compensation method illustrated in Figure 6(c) can be employed. Here, the flats A and C are supported by equivalent clamping methods, which induce roughly equivalent deformations on their surfaces. Accordingly, this significantly reduces the extent to which the optical path length differences illustrated in Figures 6(a) and 6(b) differ, thereby reducing the influence of the clamping deformation on the measurement results.
When the materials and/or dimensions of the flats differ, the mechanical compensation method will present uncertainty and cannot be applied with reasonable success. erefore, the standard measurement configuration must be employed with the bottom flats placed directly below on the planar surface as usual and the top flats subject to gravityinduced deformation. However, in the proposed method, only flats A and C s are placed in the top position. erefore, only these flats introduce clamping deformation into the measurements. Accordingly, the wavefront expressions in equation (3) can be revised as follows: Here, w a (x, y) represents the wavefront due to the deformation of flat A and w c (x, y) represents wavefront due to the deformation of flat C s . ese deformation wavefronts can be calculated for flats with various materials, dimensions, thicknesses, and clamping mechanisms and assembled into a deformation wavefront database. e present work assembled a deformation wavefront database based on deformation calculations using a thin plate model and finite element simulations. en, the calculated values of w a (x, y) and w c (x, y) can be subtracted from the measured wavefront W s1 -W s5 to obtain reasonable approximations of the ideal wavefront expressions givens in equation (3). Furthermore, the wavefronts W s1 -W s5 , respectively, represent the new wavefronts obtained from each combination of interference fringes captured during the absolute measurement process, which contain deformation data. International Journal of Optics e present work therefore provides a mechanical clamping structure combined with a computational method to compensate for the gravity-induced deformations of the three flats in the multiposition rotation absolute measurements when the materials and dimensions are equivalent. Naturally, the computational compensation method is fully applicable as well when the materials and dimensions of the flats are different.

Results and Analysis of Absolute Surface Measurement
Experiments.
e surface reconstruction performance of the proposed multiposition rotation absolute measurement method in conjunction with the corresponding deformation compensation was evaluated experimentally using three reference mirrors fabricated with equivalent materials, dimensions, and surfaces having an average PV deviation of about 20 nm by theoretical estimation. In addition, the same clamping mechanism structure was applied to all mirrors during the experiments, where the frames were only glued on their sides to ensure consistent clamping deformations according to Figure 6. e experiments used a vertical Fizeau wavelength-tuned interferometer with a common-path design, an aperture of 300 mm, and a variable frequency laser with a power of 30 mW and a center wavelength of 688 nm. Multiposition rotation absolute measurements were obtained with N � 10, and the effect of random errors on the measurement results were reduced by basing each measurement result on an average of ten measurement data.
First, we evaluated the surface reconstruction performance of the proposed computational method to compensate for the gravity-induced deformations of reference mirrors A and C in the multiposition rotation absolute measurements. Accordingly, interferometric wavefront measurements W s1 (x, y), W s2 (x, y), W s3 (x, y), W s4 (x, y), and W s5 (x, y) were obtained. Substituting the average values into equation (6)    Next, we evaluated the surface reconstruction performance of the proposed mechanical clamping combined with computational method applied to reference mirror C in the multiposition rotation absolute measurements. Accordingly, interferometric wavefront measurements W 1 (x, y), W 2 (x, y), W 3 (x, y), W 4 (x, y), and W 5 (x, y) were obtained. Substituting the average values into equation (3) yields the reconstructed surfaces of reference mirrors A, B, and C shown in Figures 8(a)-8(c), respectively. e average PV deviations from a perfect plane obtained for the three mirrors were 11.70 nm, 12.41 nm, and 11.25 nm, respectively.
Comparing the results of the compensation methods with the reconstructed surfaces in Figure 4, we note that the maximum PV deviation was 1.7 nm from a perfect plane obtained for mirrors A, B, and C by the mechanical clamping combined with the computational compensation method.
e results indicate that the combination compensation method provides the high-precision surface reconstruction outcomes.

Conclusion
e present work addressed the limitations of past work by proposing a modified three-flat testing method based on multiposition rotation interference absolute surface measurement technology that combines the basic theory of N-position rotation with the separability of surface wavefront functions into sums of even and odd functions. In addition, we proposed a mechanical clamping combined with computational method to compensate for the gravityinduced deformations of the three flats in the multiposition rotation absolute measurements when the materials and dimensions of the flats are equivalent. e results of simulations demonstrated that the average PV differences between mathematically generated irregular surfaces and the surfaces reconstructed by the proposed multiposition rotation absolute measurement method are less than 0.1 nm. e results of laboratory experiments demonstrated that the deformation compensation method provides sufficiently high-precision surface reconstruction outcomes. Accordingly, we conclude that the present work provides an important contribution to the field of interferometry measurements of large aperture, high-precision optical component surfaces.

Data Availability
e simulation analysis data and experimental measurement data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.