Interferometric stitching method for testing cylindrical surfaces with large apertures

Shuai Xue; Shuai Xue; Shuai Xue; Shuai Xue; Yifan Dai; Yifan Dai; Yifan Dai; Shengyue Zeng; Shanyong Chen; Shanyong Chen; Shanyong Chen; Ye Tian; Ye Tian; Ye Tian; Feng Shi; Feng Shi; Feng Shi

doi:10.1364/OE.428713

1. Introduction

Cylindrical surfaces have different radii of curvature (Roc) in the x- and y-directions, and therefore, they are widely used for correcting astigmatism in laser resonators [1,2] and the beam shaping units [3] of high-energy laser systems. Cylindrical surfaces need to be large for achieving performance improvements such as higher energy and better focus; this implies an increase in the angular aperture and the dimensions of the surface. To the best of the authors’ knowledge, cylindrical surfaces with dimensions larger than 0.5 m and angular apertures larger than R/3 are already required by high-energy laser systems. The typical demand for the surface figure accuracy of these surfaces is better than 0.1 µm; deterministic figuring is required to meet this requirement. The surface figure of the cylindrical surfaces must be measured beforehand to guide the iterative fabrication process successfully. As the commonly used high-accuracy testing method, interferometry requires a customized computer-generated hologram (CGH) [4–6] to transform standard flat or spherical wavefronts into cylindrical wavefronts with the same shape as the test surface. However, the state-of-the-art manufacturing technologies can create CGHs that are limited in terms of dimensions (120 × 120 mm²) and F/# (typically F/6). The dimension and the angular aperture of large cylindrical surfaces could be both beyond the ability of CGHs. Therefore, the surface figure tests of cylindrical surfaces with large dimensions and angular apertures have become a bottleneck in the development of high-energy laser systems.

Peng et al. and Chen et al. performed excellent work on the stitching method for cylindrical surfaces with large angular apertures. The misalignment aberration caused by the stitching motion [7–9] during cylindrical surface testing was investigated by Peng et al. and Hou et al. The stitching algorithm for testing cylindrical surfaces with large angular apertures was developed [10–14] by Peng et al. and Chen et al. However, the dimension of the test cylindrical surface was only 53 × 50.8 mm² (linear direction × arc direction) [10]. Dimensions in the linear direction are within the dimensions of the CGHs. The stitching was conducted only along the arc direction for using an F/1.5 CGH to test the cylindrical surface with R/0.5. Liu et al. investigated the use of a cylindrical lens and a Schmidt-like corrector plate to test a mandrel (near-cylinder optics) [15,16]. The surface figure error in the local region of the mandrel was tested successfully. Remarkable work has been conducted by Liu et al.; however, the stitching method for obtaining the full aperture surface figure is yet to be researched. Further, the dimension of the mandrel in the linear direction is ∼50 mm, as estimated from the figure of the published Ref. [15]; this is considerably smaller than 0.5 m. Ma et al. reported an absolute test method for cylindrical surfaces using the conjugate differential method [17]. A higher test accuracy was achieved for the testing cylinders; however, the research goal was not to test cylinders with large apertures. The aperture of the cylindrical surface for validation was only approximately 40 × 40 mm² [17]. The dimensions in the linear direction and the aperture angle are both within the range of the CGHs. Reardon et al. investigated the absolute interferometric test of a cylindrical surface with a fiber optic [18,19]. The aperture of the tested cylindrical optics was 40 mm, which was far from the semi-meter class.

Although all the aforementioned studies focused on testing cylindrical surfaces, stitching along only the arc direction of cylindrical surfaces can be performed via the stitching algorithm and motion stages; moreover, only cylindrical surfaces with large angular apertures can be adapted. The largest reported dimensions of tested cylindrical surfaces are only approximately 50 × 50 mm², which are far smaller than those (typically 400 × 400 mm²) required in high-energy laser systems. The determination of overlapping point pairs between subapertures for cylindrical surfaces during two-dimensional stitching is the key issue faced when developing the stitching algorithm. In addition, the serialization design and motion system modeling of CGHs are key issues faced when establishing the two-dimensional stitching motion system. Moreover, the testability of cylindrical surfaces has not been clearly identified. There exists a slight lack of knowledge regarding designing of the shape parameters of the cylindrical surfaces, among engineers designing and constructing high-energy systems. Moreover, these engineers may hesitate in adopting cylindrical surfaces with large apertures that could enhance the performance of the systems. The insufficient research on the testability of cylindrical surfaces hinders the development of high-energy laser systems to some extent. Thus, the development of an interferometric method, whereby stitching tests can be performed along both the linear and arc directions for cylindrical surfaces with large apertures, and thorough analyses of the testability remain challenging.

To this end, we propose an interferometric stitching method for cylindrical surfaces with large apertures to overcome the limited testability in the case of conventional methods. The proposed method allows for a high large aperture (both in terms of the dimension and angular aperture) adaptive capacity; this is achieved via the development of a subaperture stitching algorithm and an interferometric stitching workstation that can adapt stitching tests along both the linear and arc directions for cylindrical surfaces. The overlapping point pairs between subapertures arranged at different two-dimensional transverse positions are determined based on ray tracing and computer graphics. The stitching workstation is equipped with a series of CGHs with different F/# for enhancing the testability, as well as a six-axis movement platform to perform two-dimensional stitching. Furthermore, the testability of the proposed method is analyzed theoretically considering the stroke of the motion system and CGH parameters. The results of the theoretical analysis indicate that the stitching workstation can adapt convex and concave cylindrical surfaces with R/# values as high as R/0.5 (semi-circular cylinder) and apertures up to 700 mm. A convex cylindrical surface with a radius of 4315 mm and an aperture of 350 × 380 mm² (arc direction × linear direction) is tested to validate the feasibility of the proposed method. To the best of our knowledge, the test cylindrical surface used for validation is the largest cylindrical surface to have been reported for surface figure interferometric tests. The proposed method can significantly enlarge the aperture of testable cylindrical surfaces, particularly those having both large dimensions and angular apertures, which cannot be tested via state-of- the-art interferometry. Furthermore, the proposed method can serve as a good reference for engineers who intend to use large cylindrical surfaces in high-energy laser systems.

The remainder of the manuscript is organized as follows. Section 2 presents the stitching algorithm and the interferometric stitching workstation that can adapt stitching tests along both the linear and arc directions for cylindrical surfaces. A theoretical analysis on the testability of the proposed method is also presented. Section 3 describes the experimental validation, and discussions are included in Section 4. Finally, the paper is concluded in Section 5.

2. Principle

2.1 Algorithm for the subaperture stitching of cylindrical surfaces along both the linear and arc directions

The basic principle of a regular stitching algorithm is minimizing height differences between overlapping point pairs of different subapertures [20]. The two key points of stitching algorithm involve finding the overlapping point pairs and minimizing height differences in terms of least-squares.

The acquired height data were presented in the imaging coordinate frame. The pixel coordinates require to be transformed into global (workpiece) transverse coordinates to determine the overlapping point pairs. For flat surfaces, a linear mapping relationship exists between the transverse coordinates at the image plane and the local coordinate frame. The transformation from the local coordinate frames located at the center of the subapertures to the global coordinate frame is also straightforward. Therefore, determining overlapping point pairs for flat surfaces is relatively simple. In contrast, cylindrical surfaces have different curvatures along the linear and arc directions, which introduces a nonlinear mapping relationship between the pixel coordinates (u, v) and the local transverse coordinates (X₀, Y₀) during testing. It is difficult to determine this relationship using theoretical geometrical optics analysis owing to the diffraction effect of the CGH. Further, the relative pose and position between the transmissive flat (TF), CGH, and the test surface is complicated; this exacerbates the nonlinear mapping effect. Moreover, because stitching is performed along both the linear and arc directions, the transformation from the transverse coordinates at the local coordinate frames to those at the global coordinate frames is more complicated. The following text will analyze the developed method for finding overlapping point pairs of subapertures in cylindrical surface testing.

We denote the two sets of measured height data of subapertures i and k as m_i = { (u_j_,_i, v_j_,_i, φ_j_,_i), j = 1, 2, …, N_i} and m_k = { (u_j_,_k, v_j_,_k, φ_j_,_k), j = 1, 2, …, N_k}, respectively, where φ_j,i denotes the measured height on the imaging pixel (u_j,i, v_j,i) of subaperture i; φ_j,k denotes the measured height on imaging pixel (u_j,k, v_j,k) of subaperture k. Further, N_i and N_k represent the total number of measurement points of subapertures i and k, respectively. The test geometry is shown in Fig. 1. The local coordinate frame (X₀, Y₀, Z₀) was built at the center of the subapertures. The global coordinate frame or workpiece coordinate frame (X, Y, Z) is built at the center of the test surface. The imaging frame or CCD frame (u, v, φ) is built at the top-left corner of the CCD. The overlapping point pairs of the subapertures are found in the global coordinate frame, i.e., the workpiece coordinate frame (X, Y, Z). However, the test data obtained by the interferometer are in the imaging frame (u, v, φ), and therefore, it is essential to establish a mapping relationship between the global coordinate frame and the imaging coordinate frame.

Fig. 1. Test geometry for cylindrical surfaces with large aperture (a) from top view, and (b) from right-side view.

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The transverse coordinates (u, v, φ) at the imaging frame and (x₀, y₀, z₀) at the local coordinate frame can be expressed as

(1)$$({{x_0},{y_0}} )\, = \,f({u,v} ),$$

where f denotes the mapping function between (u, v) and (x₀, y₀).

One method to obtain f is pasting an array of fiducials on the test surface [21]. f can be obtained by experimentally determining the transverse coordinates on both the test surface and imaging plane. However, it is difficult to acquire high accuracy using the fiducials method because it is limited by the accuracy of finding transverse coordinates on these two surfaces. Further, this method has a limited resolution because of the number of fiducials. To obtain the mapping function f with relatively high accuracy and resolution, ray tracing using macro programming in optical design software (e.g., Zemax) is introduced. The ray-tracing technique has been widely used in retrace error correction [22–25] and mapping distortion correction [26].

First, an optical model of the test system was established, and then, an array (e.g., 512 × 512) of rays was traced from the imaging plane to the test surface. The ray tracing result presented in the form of four columns of data is saved as a ray-tracing look-up table. The ray tracing look-up table is a numerical form of f. The transverse coordinates (u, v) of any ray on the imaging plane can help determine the corresponding coordinates (x₀, y₀) on the test surface via interpolation. As an example, consider a concave cylindrical surface with an aperture of 300 mm × 300 mm and Roc of 2900 mm. A cylindrical CGH with F/9.6, focus distance of 1012.3 mm, and aperture of 105 mm ×100 mm is utilized to test the surface. The optical model of the test system is established using Zemax, as shown in Fig. 2. The footprints of the rays on the imaging plane and on the test surface are shown in Fig. 3. Nonlinear mapping exists between the imaging plane and the test surface. The rays were traced from the imaging plane to the test surface by macro programming in Zemax. Thus, the numerical form of f is obtained.

Fig. 2. Optical model of the test system for the cylindrical surface example.

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Fig. 3. Footprints of rays on the imaging plane (blue) and that on the test surface (red).

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Determining the approach for establishing the mapping relation between the global coordinate frame (X, Y, Z) and the local coordinate frame (X₀, Y₀, Z₀) is a typical computer graphics issue.

Let (x₀, y₀, z₀) be a point on the nominal test surface. Then,

(2)$${z_0} = \sqrt {{r^2} - x_0^2 - y_0^2} - r, $$

where r denotes the Roc of the cylindrical surface.

Without loss of generality, for the off-axis subaperture i with the off-axis angle α_i in the arc direction and the off-axis distance l_i in the linear direction, coordinates (x, y, z) in the global frame can be related to those in the local frame by rigid body transformations based on computer graphics, i.e.,

(3)$${[x\textrm{ }y\textrm{ }z\textrm{ 1}]^\textrm{T}} = {T_y}{T_{r^{\prime}}}{R_B}{T_r}{[{x_0}\textrm{ }{y_0}\textrm{ }{z_0}\textrm{ 1}]^\textrm{T}}, $$

where

\begin{aligned} &{T_y} = \left[ {\begin{array}{{cccc}} 1&0&0&0\\ 0&1&0&{{l_i}}\\ 0&0&1&0\\ 0&0&0&1 \end{array}} \right] {T_{r^{\prime}}} = \left[ {\begin{array}{{cccc}} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&{ - r}\\ 0&0&0&1 \end{array}} \right], {R_B} = \left[ {\begin{array}{{cccc}} {\cos \alpha }&0&{\sin \alpha }&0\\ 0&1&0&0\\ { - \sin \alpha }&0&{\cos \alpha }&0\\ 0&0&0&1 \end{array}} \right], \;\textrm{and}\;\;\\ & {T_r} = \left[ {\begin{array}{{cccc}} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&r\\ 0&0&0&1 \end{array}} \right]\end{aligned}

According to Eqs. (1), (2), and (3), the acquired height data m_i = { (u_j_,_i, v_j_,_i, φ_j_,_i), j = 1, 2, …, N_i} and m_k = { (u_j_,_k, v_j_,_k, φ_j_,_k), j = 1, 2, …, N_k} of subaperture i and subaperture k in the pixel frame can be transformed into w_i = { (x_j_,_i, y_j_,_i, φ_j_,_i), j = 1, 2, …, N_i} and w_k= { (x_j_,_k, y_j_,_k, φ_j_,_k), j = 1, 2, …, N_k} in the global coordinate frame. Then, the projections of w_i and w_k on the OXY plane can be used to find the overlapping point pairs through the Delaunay triangulation and the convex hull [27].

After the overlapping point pairs are identified, ${\varphi _{{j_o}}}_{,i}\;\;\;{\varphi _{{j_o}}}_{,k}$ (j_o= 1, 2, …, N_o) is used to represent the measured height in the overlapping region between subapertures i and k, where N_o denotes the number of points in the overlapping region of subapertures i and k. Because of the misalignments of the stitching motion for testing different subapertures, the measured height data φ are composed of the actual surface figure error δ, misalignment aberrations, and data positioning errors. Liu et al. [16] and Peng et al. [7] have deduced the mathematical models for the misalignment aberration, which described relationship between misalignment aberrations and possible adjustment errors based on the first-order approximate principle. The low-order misalignment aberrations for testing the cylindrical surfaces are piston, x-tilt, y-tilt, y-power, and twist [7]. Golini et al. [28] have deduced the aberrations induced by positioning errors, which comprises the x-shift, y-shift, and clock. Therefore,

(4)$$\begin{aligned} {\varphi _{{j_\textrm{o}},i}} &= {\delta _{{j_\textrm{o}}}}_{,i} + {a_i} + {b_i}{x_{{j_\textrm{o}},i}} + {c_i}{y_{{j_\textrm{o}},i}} + {d_i}y_{{j_\textrm{o}},i}^2 + {e_i}{x_{{j_\textrm{o}},i}}{y_{{j_\textrm{o}},i}}\\ &+ t_i^x\frac{{\partial {\delta _{{j_\textrm{o}},i}}}}{{\partial {x_{{j_\textrm{o}},i}}}} + t_i^y\frac{{\partial {\delta _{{j_\textrm{o}},i}}}}{{\partial {y_{{j_\textrm{o}},i}}}} + {\theta _i}[{x_{{j_\textrm{o}},i}}\frac{{\partial {\delta _{{j_\textrm{o}},i}}}}{{\partial {y_{{j_\textrm{o}},i}}}} - {y_{{j_\textrm{o}},i}}\frac{{\partial {\delta _{{j_\textrm{o}},i}}}}{{\partial {x_{{j_\textrm{o}},i}}}}] \end{aligned}, $$

(5)$$\begin{aligned} {\varphi _{{j_\textrm{o}},k}} &= {\delta _{{j_\textrm{o}},k}} + {a_k} + {b_k}{x_{{j_\textrm{o}},k}} + {c_k}{y_{{j_\textrm{o}},k}} + {d_k}y_{{j_\textrm{o}},k}^2 + {e_k}{x_{{j_\textrm{o}},k}}{y_{{j_\textrm{o}},k}}\\ &+ t_k^x\frac{{\partial {\delta _{{j_\textrm{o}},k}}}}{{\partial {x_{{j_\textrm{o}},k}}}} + t_k^y\frac{{\partial {\delta _{{j_\textrm{o}},k}}}}{{\partial {y_{{j_\textrm{o}},k}}}} + {\theta _k}[{x_{{j_\textrm{o}},k}}\frac{{\partial {\delta _{{j_\textrm{o}},k}}}}{{\partial {y_{{j_\textrm{o}},k}}}} - {y_{{j_\textrm{o}},k}}\frac{{\partial {\delta _{{j_\textrm{o}},k}}}}{{\partial {x_{{j_\textrm{o}},k}}}}] \end{aligned}, $$

where a_i, b_i, c_i, d_i, e_i, $t_i^x$, $t_i^y$, and θ_i denote the coefficients of the piston, x-tilt, y-tilt, y-power, twist, x-shift, y-shift, and clock of subaperture i, respectively. Further, a_k, b_k, c_k, d_k, e_k, $t_k^x$, $t_k^y$, and θ_k represent the coefficients of the piston, x-tilt, y-tilt, y-power, twist, x-shift, y-shift, and clock of subaperture k, respectively. Notably, the misalignment induced aberrations presented in Eqs. (4) and (5) are all low-order aberrations. When the surface figure of the test surface is large or the test surface is seriously misaligned, a nonzero fringe pattern might be obtained. As a result, high-order misalignment aberrations (e.g., high-order coma) will be introduced into the measurement result [8]. The types and mathematical descriptions of high-order misalignment aberrations have been analyzed by Peng et al. There exists a proportional relation between coefficients of high-order and low-order misalignment aberrations. Therefore, the high-order misalignment aberrations can be separated from the measurement results when the coefficients of the low-order misalignment aberrations are determined [8].

The height differences between φ_j,_i and φ_j,_k in the overlapping region should be minimized in the least-squares (LS) sense, that is,

(6)$$\min F = \sum\limits_{k = 1}^s {\sum\limits_{i = 1}^s {\sum\limits_{{j_o} = 1}^{{N_o}} {({\varphi _{{j_o}}}_{,i} - {\varphi _{{j_o}}}_{,k})} } }, $$

where s denotes the number of subapertures. By solving Eq. (6), the coefficients of the misalignment aberrations and data positioning errors for all subapertures can be acquired simultaneously. Then, the calculated misalignment aberrations and data positioning errors can be removed from all subapertures, and the stitching can be completed. After the subapertures are stitched using the algorithm proposed by the authors, the incorrect aberrations induced by misalignment Δϕ can be coupled with the stitching result ϕ. Then the misalignment aberration Δϕ can be estimated in the least-squares (LS) sense, that is,

(7)$$\min F = \sum\limits_{\textrm{i} = 1}^{{N_f}} {({\phi _\textrm{i}} - \Delta {\phi _\textrm{i}}} ), $$

(8)$$\Delta {\phi _\textrm{i}} = {A_i} + {B_i}{x_i} + {C_i}{y_i} + {D_i}y_i^2 + {E_i}{x_i}{y_i}. $$

where A_i, B_i, C_i, D_i, and E_i denote the coefficients of the piston, x-tilt, y-tilt, y-power, and twist of the full aperture. N_f denotes the total number of measurement points of the full aperture. Then the misalignment aberration Δϕ is subtracted from the stitching result ϕ to separate the misalignment aberration.

2.2 Interferometric stitching workstation performing stitching along both the linear and arc directions for cylindrical surfaces with a large aperture

Using the proposed stitching algorithm, the subapertures for cylindrical surfaces with large dimensions and angular apertures can be stitched theoretically. However, for practical measurements, issues regarding the stitching motion system, CGH design, and evaluation of the measurement ability need to be addressed. These techniques are analyzed as follows.

2.2.1 Stitching motion system

As shown in Fig. 4, stitching is conducted along the linear (y axis) and arc (B axis) directions for cylindrical surfaces with large dimensions and angular apertures. Therefore, it requires translational and rotational motion along and around the axis of rotation (focal line) with a high positioning accuracy. The cylindrical surfaces should be rotated around the focal line; however, this is difficult in practice, especially for cylinders with a large radius of curvature (e.g., 2000 mm). For the sake of convenience, the axis of the rotary table is located near the cylindrical test surface. Hence, two orthogonal translations along the x-and z-axes with tolerable positioning errors are required. Further, tilting around the optical axis (z-axis) to remove twist and pitching to remove y-tilt are necessary for aligning the test surface to the CGH. The stitching motion system requires a 4-axis (x, y, z, and B) electric control motion system and a 2-axis (A and C) manual (or electric controlling) adjusting table.

Fig. 4. Freedom of motion requirements for testing cylindrical surfaces with large dimension and angular aperture.

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Based on the position and pose of the interferometer and the test surfaces, there are three typical configurations for the stitching motion system: vertical, horizontal with the horizontal axis of the cylindrical surfaces, and horizontal with the vertical axis of the cylindrical surfaces. In the vertical configuration, the optical axis of the interferometer is vertical, and the test surface lies on the platform towards the interferometer. The most attractive merit of this configuration lies in its easier clamping and support for large and heavy test surfaces. The disadvantage is that many cylindrical surfaces (e.g., concave cylindrical surfaces with large Roc) cannot be tested owing to the maximum length of the interference cavity. The horizontal configuration implies that the optical axis of the interferometer is horizontal; this configuration can adapt to more cylindrical surfaces because the space is less restricted. The linear direction of the test surface can be either horizontal or vertical. Because of the convenient arrangement of the rotary stage for rotating around the focal line, a horizontal configuration with the vertical axis of cylindrical surfaces is preferred.

As shown in Fig. 5, a stitching motion system is established based on the requirements of movement, adjustment, and preferred configuration. The motion system comprises four electric control motion axes (x, y, z, and B) and a two-axis manual adjusting table (A and C). The stokes of the x-axis, y-axis, and z-axis are 700 mm, 700 mm, and 100 mm, respectively. The positioning accuracy for each translation axis is approximately 30 µm. The range of the rotary stage (B) is 360°; the motion resolution and repeated positioning accuracy are 0.00125° and 0.005°, respectively. The three translations are realized through a ball screw with a servomotor drive, and the rotary stage is realized through rotary tables with a servomotor drive. A set of Zygo GPI 6” is utilized as the interferometer, and the CGH is located between the interferometer and the test surface. The stitching workstation is placed on an air-floating platform.

Fig. 5. Stitching motion system for testing cylindrical surfaces with large dimension and angular aperture.

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2.2.2 CGH design

A cylindrical CGH is utilized to convert a standard wavefront (e.g., flat or spherical wavefront) into a cylindrical wavefront. A cylindrical CGH can be used for concave and convex cylindrical surfaces with different curvatures. Therefore, the CGH is considered the “gold standard” for testing cylindrical surfaces. Therefore, a good CGH design forms the basis for acquiring accurate measurement results. The design method of CGH for cylindrical surfaces is similar to that of CGH designed for aspheric surfaces [29]; the method is explained below.

Without loss of generality, a convex circular cylindrical surface with an aperture of approximately 380 mm (diameter) and Roc of 4315 mm is used as an example for designing CGH with cylindrical surfaces. The test surface parameters are illustrated in Fig. 6. The aperture of the CGH is smaller than 6”, which is limited by the state of the lithography process. The region for measurement needs to be slightly smaller than 6” for the edge of the CGH to be used as the align region. For this demonstration, the aperture of the testing region is set to 130 mm.

Fig. 6. Parameters of the test surface.

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The single-pass model shown in Fig. 7 is established for better optimization of the CGH phases. The surface type of the test cylindrical surface is set as biconic. The refractive index of the medium before the test cylindrical surface is set to zero, and hence, the emergent beam is forced to be perpendicular to the shape of the test surface based on Snell’s law. Apparently, for a fixed CGH aperture, the testable aperture of the test surface is larger with a decreasing distance from the test surface to the CGH. However, a space that is too narrow is not practical for the adjustment. The distance from the test surface to the CGH is set to 100 mm to install the adjustment platform for the test surface and the CGH. The material of the CGH is silica, and the thickness is set to 6.35 mm. The second surface of the CGH is set as the phase surface with a surface type of Zernike standard phase [30], which is defined as

(9)$$\mathrm{\Phi }\textrm{ = }M\sum\limits_{i = 1}^N {2\pi } {A_i}{Z_i}(\rho ,\varphi ), $$

where N denotes the number of Zernike coefficients in the series, A_i represents the coefficient of the ith Zernike standard polynomial, ρ denotes the normalized radial ray coordinate, φ denotes the angular ray coordinate, and M denotes the diffraction order. Coefficient A_i has units of waves; one wave is 2π radians. A +1-order diffraction beam is utilized as the test beam (i.e., M = 1), and a tilt carrier is applied to the CGH to separate the diffraction beams of other orders. The incident beam is inclined relative to the optical axis; the tilt angle of the incident beam is set to 2.4° around the x-axis. A paraxial surface with a focal length of 200 mm is used to simulate the ideal image lens.

Fig. 7. Single pass model of the test system.

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The incident beam is inclined by adding a coordinate beak before the paraxial surface in the model. Phase optimization is performed after the model is established. The maximum term of the Zernike standard surface is set to 22; i.e., N = 22. Further, Z₁= Z₂= 0, and Z₃–Z₂₂ denote the optimization variables. The optimization goal is to minimize the root-mean-square (RMS) value of the wavefront focused on by the paraxial surface. The contour map of the CGH phase after optimization is shown in Fig. 8. The contour spacing is 500 periods. Figure 8 indicates that the average spacing is estimated to be 130 mm/17/500 = 15 µm; a line spacing of 15 µm is relatively sparse, which is helpful for manufacturing CGH with high accuracy. Because the carrier is tilt along the arc direction, the contour lines are all straight lines. This will be beneficial to guarantee the manufacturing accuracy of the CGH.

Fig. 8. Contour map of the CGH phase.

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According to the reversibility of light, a double-pass model as shown in Fig. 9 is established based on the optimized single-pass model. In the double-pass model, the incident beam is collimated with a field angle of −2.4° around the x-axis. The parameters of the CGH are the same as those of the single-pass model. The collimated beam is converted into a cylindrical wavefront by the CGH. Then, the test beam is reflected from the test surface perpendicularly; the test beam is then converted into a collimated beam after being diffracted by the CGH. The collimated beam is transformed into an ideal image using the paraxial surface. A coordinate break with a 2.4° tilt about x is added before the paraxial surface.

Fig. 9. Double pass model of the test system.

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The footprint of the test beam on the test surface is shown in Fig. 10. This shows that the test aperture is far smaller than that of the test surface. Stitching is required along both the linear and arc directions. The detailed stitching scheme is presented in the following experiment. Figure 11 shows the residual aberration with peak-to-valley (PV) 0.0023λ (λ=632.8 nm) and RMS 0.0002λ. The design accuracy meets the test accuracy requirements.

Fig. 10. Footprint of the test beam on the test surface.

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Fig. 11. Residual aberration.

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Avoiding interference from other diffraction orders is essential for obtaining good results [31–33]. Multiple configurations with different diffraction orders in the double-pass model were set to guarantee good isolation of the +1 order diffraction beam (double pass). Considering that the diffraction efficiency decreases very rapidly with higher diffraction orders, the orders of (+1, −1), (+1, 0), (+1, +1), (+1, +3), (−1, −1), (−1, 0), (−1, +3), (0, 0), (0, +3), (+3, −1), (+3, 0), and (+3, +3) were set to determine whether there are any incorrect diffraction orders. A pinhole with a diameter of 2 mm was placed at the image plane. Figure 12 shows a spot diagram of all the configurations at the image plane. No other lights except the (+1, +1) beam (blue) can arrive at the image plane.

Fig. 12. Spot diagram of all configurations at the image plane.

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The outer ring (R = 65–70 mm) of the CGH is set as the alignment region for aligning the CGH and the interferometer. The design method of the phases for the alignment region is similar to that of the testing region; however, the optimization variable is only Z3. Finally, the CGH phases are encoded into the GDSII format for manufacturing [34].

Following the above process, a series of CGHs with F/2.8, F/5.8, F/9.6, and F/34 were designed and manufactured. The parameters of the CGHs are listed in Table 1. The two manufactured CGHs are illustrated in Fig. 13.

Fig. 13. Manufactured CGH of (a) F/34 (b) F/2.8.

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Table 1. Parameters of the CGH for the cylindrical surfaces (Units: mm)

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2.2.3 Evaluation of measurement ability

A convex or concave cylindrical surface could be tested by placing its rotation axis coincident with the focus line of the CGH. The convex cylindrical surfaces are tested in the converging portion of the measurement beam; the concave surfaces are tested in the diverging portion of the beam. The cone of the beam from the CGH needs to be faster than the test surface to test the entire surface in one shot. In other words, the F/# of the CGH must be smaller than the R/# of the test cylindrical surface. Moreover, the dimensions of the test cylindrical surface along the linear direction must be smaller than that of the CGH. For convex surfaces, the radius of the test surface must be no longer than the focal length of the CGH. However, the CGH usually has limited F/# and dimensions along both the linear and arc directions because of the limitations of its manufacturing technology. Stitching along the linear or/and arc directions is required for cylindrical surfaces with large dimensions and/or angular apertures to obtain full aperture test results. Thus, the measurement ability for cylindrical surfaces is limited by the available CGH state of art, stitching motion system, and stitching algorithm. A stitching workstation for a cylindrical surface was established by incorporating the aforementioned stitching algorithm, stitching motion system, and available CGHs. The measurement abilities of the cylindrical stitching workstation for concave and convex cylindrical surfaces are illustrated in Figs. 14 and 15, respectively. In these figures, D_A denotes the dimension (diameter) along the arc direction of the test cylindrical surfaces. D_L denotes the dimension (diameter) along the linear direction of the cylindrical surfaces.

Fig. 14. Concave cylindrical surface test ability.

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Fig. 15. Convex cylindrical surface test ability.

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As shown in Fig. 14, concave surfaces with parameter coordinates (R, D_A) below the line R/D_A = 2.8, R/D_A = 5.8, R/D_A = 9.6, and R/D_A = 34 can be tested in one shot using CGH with F/2.8, F/5.8, F/9.6, and F/34, respectively. Stitching along the arc direction is required for surfaces with parameter coordinates (R, D_A) above the line R/D_A = 2.8. The cylindrical surfaces with parameter coordinates (R, D_A) above the red line noted by R/0.5 are super semi-cylinders, which are unusual and excluded in this study. Stitching along the linear direction is required for surfaces with D_L larger than 130 mm (for F/34 CGH, 120 mm, 105 mm, and 100 mm, for F/2.8, F/5.8, and F/9.6, respectively).

The measurement ability of the convex cylindrical surfaces is shown in Fig. 15. Consider F/2.8 as an example. Convex surfaces with parameter coordinates (R, D_A) located below the line R/ D_A = 2.8 and the left side of line R = 336 mm can be tested in one shot by utilizing CGH with F/2.8. Convex surfaces with parameter coordinates (R, D_A) located above the line R/DA = 2.8, and the left side of line R = 336 mm can be tested by stitching along the arc direction using CGH with F/2.8. The surfaces with D_L larger than 130 mm can be tested by stitching along the linear direction using CGH with F/2.8. Cylindrical surfaces with parameter coordinates (R, D_A) above the red line noted by R/0.5 are convex super hemi-cylinders, and they are unusual and excluded in this study. For a given cylindrical surface, Fig. 15 can be a good reference for selecting the appropriate CGH. If a cylindrical surface can be tested by more than one CGH, the F/# of the most appropriate CGH should be nearest to the R/# of testing the surface. If F/# is far smaller than R/#, the test surface will appear too smaller on the interferometer monitor and the full resolution of the monitor is not fully utilized to resolve the fringes; and the resolution of the measured data will be lower. The percentage of full-size can be calculated as the ratio of the F/# to R/#. For example, F/5.8 and R/10 will result in an interferogram that is 58% of full size. For a cylindrical surface with roc of 4000 mm, and diameter of 100 mm, CGH with F/34, F/9.6, F/5.8, and F/2.8 can be used to test the surface. Among them, F/34 is the most appropriate CGH since the it will result in an interferogram that is 85% of full size. In comparison, the percentage value is 24%,14.5%, and 7%, for F/9.6, F/5.8, and F/2.8, respectively.

Thus, convex and concave cylindrical surfaces with dimensions up to 700 mm along both linear and arc directions and angular apertures up to R/0.5 can be tested theoretically by utilizing the developed stitching algorithm, stitching motion system, and CGHs.

3. Experiments

A very broad range of cylindrical surfaces with large apertures was measured successfully by utilizing the proposed interferometric stitching method. The parameters of the cylindrical surfaces are listed in Table 2. The #1 cylindrical surface has the largest aperture. The detailed test process for the #1 surfaces is provided to validate the feasibility of the proposed method.

Table 2. Parameters of cylindrical surfaces tested by the proposed method (Units: mm)

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As summarized in Table 1 and Fig. 6, the #1 surface is a convex cylindrical surface. Its Roc is 4315 mm, and its aperture is 350 mm ×380 mm (arc direction × linear direction). The F/34 CGH listed in Table 2 is used to test the surface. In the linear direction, the testable aperture in one shot equals the aperture of the CGH. Indeed, 380 mm along the linear direction is far larger than that of CGH (F/34, 130 mm), and therefore, stitching along the linear direction is inevitable. The convex surfaces must be tested in a convergent beam. Therefore, in the arc direction, the testable aperture in one shot is smaller than the aperture of the CGH. Further, the aperture angle of the test surface is 4.6487°, which is larger than that of the CGH (F/34, 1.6870°). Thus, it is necessary to perform stitching along the arc direction.

The lattice design for subapertures is basically determined by the requirements of overlapping ration and full aperture covering capability with less subapertures [35]. The overlapping ration should better larger than 10%. Considering the aperture angle of the CGH and the test surface, at least four columns of subapertures are required to cover the full aperture. Considering the dimension of the CGH and the test surface along the linear direction, at least the four rows and three rows of subapertures are required for the inner two columns, and the outer two columns of subapertures, respectively. Since none of the overlapping ration or subapertures amounts requirements is to be met accurately, we propose to roughly estimate the location of subapertures by approximation. After subtle and complicated computation, the lattice design of the subapertures is shown in Fig. 16. Four columns of subapertures are arranged with β = −0.5975°, 0.5975°, 1.7926°, and −1.7926°. The number of subapertures was 16; the translation distance between the middle two columns of the subapertures is 75 mm, and for the outer two columns, it is 100 mm. The measurement path planning strategy follows the shortest path planning principle, i.e., the measurement sequence is (#13, #14, #15, #2, #1, #0, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12).

Fig. 16. Subaperture layout.

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As shown in Fig. 17, a set of Zygo 6” GPI was used as the interferometer. The test surface was mounted on the developed stitching motion system. The linear direction of the test surface was positioned along the y-axis of the stitching motion system. A 4-axis (x, y, z, and B) electric control motion system was incorporated into the stitching motion system to perform the stitching motion required by the test surface. The distance between the axis of the rotation stage and the focus line of the test cylindrical surface is l = 4265 mm. This distance is guaranteed by the location between the rotation stage and the fixture which is manufactured by ultra-precision lathe. The nominal rotation and translation amounts (absolute amounts) for all subapertures are shown in Table 3, where the translation amounts x and z can be obtained according to computer graphics, i.e.,

(10)$$x\, = \,l{-}l\cos(B ),$$

(11)$$z\, = \,\sin(B ).$$

Fig. 17. Experimental setup.

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Table 3. The nominal rotation and translation amounts for all subapertures

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The tolerable position errors should be carefully determined by the required test accuracy, the stitching algorithm, and the spatial frequency of the interferometer. For the considered test surface applied in infrared system, the required test accuracy is about 0.1 µm. The proposed algorithm uses x-shift, y-shift, and clock terms to compensate the positioning errors of subapertures [28], therefore the positioning tolerance can be relaxed. The resolution of the CCD equipped in the interferometer is 1000 pixel × 1000 pixel. The dimension along the linear direction of the test surface is 130 mm, therefore the spatial frequency of the acquired test data is 130 µm/pixel. Commonly, positioning error smaller than pixel/3 (i.e., 43 µm) is tolerable. For the experiment setup, the positioning accuracy for each translation axis is approximately 30 µm, which is within the tolerable error. The tolerable error of the positioning error can be further relaxed using an iteration stitching algorithm [27]. The iteration stitching algorithm can best estimate and then compensate the positioning errors that are allowed to be at the millimeter or degree level.

An additional two-axis (A and C) manual adjusting table was equipped to eliminate misalignment aberrations. Null fringes were acquired by fine-tuning the z, A, B, and C axes. Therefore, the misalignment aberrations of the x-power, y-tilt, x-tilt, and twist are eliminated.

Figure 18 shows the measured heights of all subapertures. The inherent wavefront distortion of the CGH is coupled with these results. The wavefront errors of CGH comprise the CGH substrate, etching depth, amplitude ratio, and pattern distortion errors [36,37]. The etching depth, amplitude ratio, and pattern distortion errors are less than 1 nm RMS, and they can be excluded. However, as the dominant error source, the CGH substrate error is approximately 10 nm RMS. Therefore, the CGH substrate error must be calibrated according to the test results. Because the substrate error influences all diffraction orders equally, the substrate error is calibrated by measuring the wavefront distortion using the zero-order diffraction from the CGH and by subtracting it from the +1-order surface measurement [37]. The calibration system is set up as shown in Fig. 19. The measured wavefront distortion using the zero-order diffraction from the CGH is shown in Fig. 20 with a PV of 0.843λ and RMS of 0.25λ. After the substrate error is calibrated, the surface figure error shown in Fig. 19 is subtracted from the substrate error shown in Fig. 20. Then, the surface figure data of all subapertures are stitched together simultaneously using the proposed stitching algorithm, as shown in Eqs. (1)–(6). The surface figure error in the full aperture is shown in Fig. 21 with a PV of 1.262λ and RMS of 0.131λ.

Fig. 18. Measured height of all subapertures.

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Fig. 19. Calibration setup.

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Fig. 20. Substrate error.

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Fig. 21. Stitching result.

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4. Discussion

When compared with state-of-the-art stitching methods for testing cylindrical surfaces, the proposed method is proved to have merits regarding the capability for adaption of larger apertures both in terms of dimensions and angular apertures. This is achieved through the development of a stitching algorithm and establishment of a stitching workstation that can perform stitching along the linear and arc directions of the cylindrical surface. In contrast, conventional methods only focused on stitching along the arc direction for cylindrical surfaces with large angular apertures. Using the proposed method combined with a series of CGH, the testability for large cylindrical surfaces is significantly enhanced. We successfully tested a convex cylindrical surface with an aperture of 350 × 380 mm², which is, to the best of our knowledge, the largest cylindrical surface to have been reported for surface figure interferometric tests. However, conventional methods only reported testing for cylindrical surfaces with an aperture of nearly 50 × 50 mm², which is approximately an order of magnitude smaller than that reported in this article. This implies that the testable aperture of cylindrical surfaces is significantly enlarged from the conventional Ф50 mm to Ф380 mm (verified by experiment, where the theoretical value is Ф700 mm) via the proposed method. Notably, this method incorporates common techniques and research results for testing of cylindrical surfaces, such as the misalignment aberration elimination technique.

Furthermore, testability of the stitching workstation based on the proposed method had been theoretically investigated, which was not reported in previous studies. By incorporating the aforementioned stitching algorithm, stitching motion system, and available CGHs, the stitching workstation can test convex and concave cylindrical surfaces with R/# values as high as R/0.5 and with apertures up to 700 mm, as indicated by Figs. 14 and 15. These figures also provide an efficient CGH selection guide and stitching strategy for a given cylindrical surface, which will prove convenient for engineers constructing high-energy laser systems to roughly evaluate the testability and the test difficulty of a cylindrical surface. For example, if a given cylindrical surface with parameters (R, D_A) located in the region noted by ‘Requires stitching along arc direction’ shown in Figs. 14 and 15, and with 120 mm < D_L < 700 mm, stitching along both the linear and arc directions is required. It is then evaluated as more difficult to test than a cylindrical surface with parameters (R, D_A) located in the region noted by ‘Single shot measurement’, and with D_L < 120 mm. Therefore, the reported method can serve as a good reference for engineers who want to use large cylindrical surfaces in high-energy laser systems. We believe that this will further promote the application of large cylindrical surfaces in devices such as high-energy lasers.

The proposed method however has a few drawbacks because many subapertures are required for testing cylindrical surfaces with large apertures. For example, the high-order aberrations introduced by the stitching motion of different subapertures will accumulate and corrupt the full aperture stitching results. Prospects are described as follows for the further development of the proposed method.

(1) Determining the nonlinear mapping between pixel coordinates (u, v) and the local transverse coordinates (X₀, Y₀) is crucial to finding the overlapping region between different subapertures. The proposed method used ray tracing from the imaging plane to the test surface to find this mapping. The accuracy of ray tracing relies on accurate system modeling. For the collimated incident beam and TF used in the proposed method, the mapping from the imaging plane to the CGH surface next to the TF was linear, hence complete system modeling could be conducted. For a spherical incident beam and transmissive sphere (TS), the mapping relation from the pixel coordinates on the imaging plane to the transverse coordinates of the aplanatic surface of the TS was not always linear. Therefore, the modeling of the TS could not be neglected in the ray tracing model. However, the optical constructures of commercial TSs are unavailable. The incomplete system modeling introduced errors to the ray tracing result. Furthermore, the resulting incorrect overlapping point pairs will lead to stitching errors. These errors will accumulate with the increasing number of subapertures for testing large cylindrical surfaces. To improve the test accuracy, an interferometer established in the laboratory with a complete system modeling should be used. Alternatively, the mapping character of the commercial TS should be identified by other methods such as the fiducials method. Further research should be conducted on large cylindrical surfaces tested using a spherical incident beam and TS before the CGH.

(2) With the increased number of subapertures, the slight aberration difference within the overlapping region between adjacent subapertures will accumulate and introduce severe errors in the stitching results. In the proposed method, the low-order misalignment aberrations, i.e., tilt, tip, x-power (or y-power), twist, x-shift, y-shift, and clock, are considered to compensate the difference in the overlapping region. However, high-order misalignment aberrations introduced to different subapertures are not negligible, especially for testing cylindrical surfaces with large angular apertures. To rectify the high-order misalignment aberration accumulation issue, a high-order misalignment elimination strategy [11] should be incorporated into the stitching process for large cylinders. Further investigation is required.

5. Conclusion

The proposed method successfully advances the stitching configuration for cylindrical surfaces from the mode of stitching only along the linear direction to that along both the linear and arc directions. Cylindrical surfaces with large dimensions and angular apertures can thus be tested. Using the proposed method, the testable aperture of cylindrical surfaces is significantly enlarged from Ф50 mm to Ф700 mm. A convex cylindrical surface with an aperture of 350 × 380 mm² was successfully tested. The proposed method will help promote the application of large cylindrical surfaces in devices such as high-energy lasers. Moreover, testability and test strategy guides are also presented. The testability and test difficulty for a given cylindrical surface can be conveniently evaluated. The proposed method can serve as a good reference for engineers who want to use large cylindrical surfaces in high-energy laser systems. However, our method also has a few drawbacks as a result of the many subapertures required for testing cylindrical surfaces with large apertures.

Funding

National Natural Science Foundation of China (U1801259, 51835013, 51875572, 51975578, 51991371); Key Research Program of Frontier Science, Chinese Academy of Sciences (XD25020317); Science Challenge Project (TZ2018006-0101-01); State Key Lab of Digital Manufacturing Equipment and Technology (DMETKF2020023); National Key Research and Development Program of China (SQ2020YFB200368-04)

Acknowledgments

The authors appreciate Mr. Lingwei Kong and Ms. Huiping Rong for their efforts in preparing the experiments for the stitching test of surfaces.

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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No.	B (°)	y (mm)	x (mm)	z (mm)
#0	−0.5975	0	44.476	0.232
#1	−0.5975	75	44.476	0.232
#2	−0.5975	150	44.476	0.232
#3	−0.5975	−75	44.476	0.232
#4	−0.5975	−150	44.476	0.232
#5	0.5975	−150	−44.476	0.232
#6	0.5975	−75	−44.476	0.232
#7	0.5975	0	−44.476	0.232
#8	0.5975	75	−44.476	0.232
#9	0.5975	150	−44.476	0.232
#10	1.7926	100	−133.416	2.087
#11	1.7926	0	−133.416	2.087
#12	1.7926	−100	−133.416	2.087
#13	−1.7926	−100	133.416	2.087
#14	−1.7926	0	133.416	2.087
#15	−1.7926	100	133.416	2.087

No.	B (°)	y (mm)	x (mm)	z (mm)
#0	−0.5975	0	44.476	0.232
#1	−0.5975	75	44.476	0.232
#2	−0.5975	150	44.476	0.232
#3	−0.5975	−75	44.476	0.232
#4	−0.5975	−150	44.476	0.232
#5	0.5975	−150	−44.476	0.232
#6	0.5975	−75	−44.476	0.232
#7	0.5975	0	−44.476	0.232
#8	0.5975	75	−44.476	0.232
#9	0.5975	150	−44.476	0.232
#10	1.7926	100	−133.416	2.087
#11	1.7926	0	−133.416	2.087
#12	1.7926	−100	−133.416	2.087
#13	−1.7926	−100	133.416	2.087
#14	−1.7926	0	133.416	2.087
#15	−1.7926	100	133.416	2.087

Interferometric stitching method for testing cylindrical surfaces with large apertures

Abstract

1. Introduction

2. Principle

2.1 Algorithm for the subaperture stitching of cylindrical surfaces along both the linear and arc directions

2.2 Interferometric stitching workstation performing stitching along both the linear and arc directions for cylindrical surfaces with a large aperture

2.2.1 Stitching motion system

2.2.2 CGH design

2.2.3 Evaluation of measurement ability

3. Experiments

4. Discussion

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (21)

Tables (3)

Equations (12)

Optics Express

F/#	Dimension (Arc direction × Linear direction)	Focus distance
F/2.8	120 × 120	336
F/5.8	120 × 105	700
F/9.6	105 × 100	1012.3
F/34	130 × 130	4415

No.	Type	Dimension (Arc direction × Linear direction)	Radius of curvature	R/#
#1	Convex	350 × 380	4315	R/12.3
#2	Convex	200 × 400	600	R/3
#3	Convex	150 × 350	450	R/3
#4	Convex	100 × 200	962.3	R/9.6
#5	Concave	200 × 400	840	R/4.2
#6	Concave	180 × 200	2900	R/16.1
#7	Concave	160 × 150	1232	R/7.7