1. Introduction

Cylindrical surfaces are widely used in optical systems, examples of which include beam shaping and anamorphic imaging [1]. Compared with cylindrical lenses, acylindrical optics (such as parabolic and elliptic cylindrical lenses) hold an advantage in that they minimize the introduction of spherical aberration, improve the performance of optical systems, and reduce the number of optics.

Typically, acylindrical surfaces not only act symmetrically in one dimension but are also slightly different from true cylinders. Therefore, regular interferometers that are equipped with transmission flat, spherical, and cylindrical nulls are not suitable for measuring acylindrical optics. While it is possible to equip any interferometer with a computer-generated hologram (CGH) so that it can adapt the wavefront to a specific acylindrical surface, this approach lacks flexibility because we must design and fabricate a new optical null for every new acylindrical surface. Over the past few decades, stitching interferometry has become a well-known method for measuring large-aperture flat [2–4], high numerical aperture (NA) spherical [5, 6], and mild aspheric surfaces [7–9]. However, it is impractical when directly applied to steep aspheres because the the steep aspheres usually contain large local slope. To get a resolvable interferogram at off-axis subaperture, we have to increase the number of subapertures [10]. Consequently, it may increase the measurement time and reduce the accuracy. Recently, the idea of variable optical nulls has been introduced to compensate for the aberration of off-axis subapertures [11, 12]. An approach that uses a partial null lens has also been reported for steep aspheric measurement [13, 14]. But these methods require additional optical elements, making the measurement system complicated and expensive. More importantly, these methods are primarily designed for rotationally sysmmetric aspheric surfaces.

Inspired by the subaperture stitching approaches incorporated with variable optical nulls [11, 12], we proposed a stitching interferometric method to measure steep acylindrical surfaces. As opposed to existing methods, auxiliary optics is not required to generate the variable aberrations. Instead, we show here that the CGH cylinder null, which is originally used to diffract an ideal cylindrical wavefront, can be used to generate a variable-aberration coma by yawing it with a small angle. Meanwhile, we demonstrate that the departure of off-axis subapertures is dominated by the cylindrical aberration coma. As a result, we can yaw the CGH cylinder null to reduce the fringe density of off-axis subapertures, making it possible to measure steep acylindrical lens with small number of subaperture. Because we only adjust the CGH by one degree of freedom, it is easy to align and calibrate the system error. This is particularly important because any movement of the variable optical null will introduce an additional aberration in the measurement.

This paper is structured as follows. First, we analyze the departure of off-axis acylindrical subapertures. Next, we explain why the aberration of off-axis subapertures can be reduced by yawing a cylinder null. To connect all subapertures together, we also propose an acylindrical stitching algorithm. Then, we demonstrate an experiment to verify the proposed method. Finally, we summarize the results obtained and conclude this paper.

2. Departure of off-axis acylindrical subapertures

First, we consider a convex acylindrical surface, as shown in Fig. 1, whose section profile is represented as

 

Fig. 1 Coordinate transformation between the off-axis subaperture and the central subaperture.

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From Eq. (2), we can solve z as a function of (y). After subtract the z coordinate of its best fitting cylinder (denoted as z_s) from z, we yield the departure of subaperture n. Reader can refer to appendix A for detail deduction. The departure of the off-axis subaperture can be described as follow,

To verify the theoretical deduction mentioned above, we carried out a simulation using ray-tracing software (Optics Software for Layout and Optimization, OSLO). The layout of the interferometric system is shown in Fig. 2, where the simulation model used a cylinder null optics (Diffraction International, H80F3C) to generate an f/3 idea cylindrical wavefront. The target to be measured is a convex acylindrical surface without any figure error. Its clear aperture is 20mm × 20mm, the conic constant is k = −1, and the ROC at the vertex is ROC = 13.984mm. Its acylindrical departure is about 81µm. Then, using the method mentioned in [15], we can calculate the nominal motion parameters for each off-axis subaperture. Following these results, we oriented each off-axis subaperture in front of the interferometric system, and obtained the subaperture data in sequence. Figure 3(a) shows the phase map of an off-axis subaperture. To characterize the aberration of the off-axis subaperture, we employed the two-dimensional (2D) Legendre polynomials to decompose the result. The symbol λ in this figure is the unit of the laser wavelength (1λ = 0.6328µm), and it is used as the scale factor in this paper to quantify the figure error. Figure 3(b) plots the coefficients of Legendre polynomials. Compared with the 2D Legendre map given in [16], it can be seen that the phase data of the off-axis subaperture is dominated by the third-order coma (the seventh term), which agrees well with the deduction mentioned above. It is therefore desirable to have an optical device that generates comas in varying amounts, enabling the conversion of a cylindrical wavefront into one that matches the departure of each subaperture.

 

Fig. 2 Sketch map of the simulation model.

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Fig. 3 Simulated phase map of off-axis subaperture (a) and its expansion coefficients by using the 2D Legendre polynomials (b).

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3. Method for reducing the fringe density of off-axis acylindrical subapertures

If the CGH cylinder null is set at normal position, the focal line of the diffracted cylindrical wavefront will intersect with the optical axis, as shown in Fig. 4(a). Instead, if the CGH cylinder null is yawed with a small angle, the focal line of the diffracted cylindrical wavefront will deviate from the optical axis, as shown in Fig. 4(b). To understand its principle, we decompose the incident plane wave into two parts. One is parallel to the surface of the CGH cylinder null, which will not pass through the CGH cylinder null. The other passes through CGH cylinder null to diffract a cylindrical wavefront. But the focal line of the diffracted cylindrical wavefront is rotated around the center of CGH with the same angle. Therefore, the aberrations induced by yawing the CGH cylinder null is similar to that when rotating the cylindrical optics around its vertex with a small angle. Actually, we have deduced a mathematical expression for the aberrations induced by rotating the cylindrical surface in [16]. We also pointed out the high-order aberrations induced by rotating the cylindrical surface is dominated by coma, and its magnitude depends on the rotation angle. On the other hand, according to the analysis mentioned in Section 2, the departure of the off-axis subaperture is dominated by coma. As such, we can reduce the fringe density of the off-axis subaperture by yawing the CGH cylinder null.

 

Fig. 4 Diffracted wavefronts when the CGH cylinder null was set at normal position (a) and was yawed with a certain angle (b).

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To verify the analysis mentioned above, we carried out another simulation using the software OSLO. Compared to the previous simulation model, the layout of the simulated model was unchanged, except that the CGH cylinder null was yawed with a small angle. Figures 5(a) and 5(d) give the simulated interferograms before and after yawing the CGH cylinder null when testing an off-axis subaperture. It can be seen that when the subaperture surface is measured without yawing the CGH cylinder null, the fringe densities of off-axis subapertures are too large to be irresolvable. While yawing the CGH cylinder null with a small angle, we can obtain an interferogram with sparse fringes. It should be noted that the interferograms given in Fig. 5 were generated based on the normal deviation between the reference wavefront and the tested surface. Because we yawed the cylinder null to generate an acylindrical wavefront, the normal deviation mentioned above became very small, resulting in a sparse interferogram. To further investigate the robustness of the proposed method, we oriented different subapertures in front of the interferometric system in sequence, then acquired the subaperture data by yawing the cylinder null with different angles. Figure 5 shows the interferograms of off-axis subapertures before and after yawing the cylinder null. These simulation results indicate that resolvable interferograms can be acquired at off-axis subapertures only after yawing the cylinder null; if not, the Nyquist condition is violated, causing the fringe densities in the outer subaperture to be irresolvable. These results also imply the incompetence of conventional stitching interferometry for steep acylindrical surfaces.

 

Fig. 5 Simulated interferograms of off-axis subapertures before (a–c) and after (d–f) yawing cylinder null when measuring acylindrical lens.

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To determine the yaw angle of CGH for each off-axis subaperture, we should calibrate the relationship between the yaw angle of CGH and the coefficient of the aberration coma in advance. This can be done by placing a cylindrical surface at the null position, then yawing the CGH with a different angle and obtaining the phase data in sequence. To eliminate the effectiveness caused by the figure error of the cylindrical surface, we can subtract the result measured at the null position from each measurement. Next, the phase difference can be expanded using the 2D Legendre polynomials. Figure 6(a) plots the expansion coefficients of the aberrations induced by yawing the CGH with 0.875°. It can be seen that only the seventh-term coma is induced, and other high-order aberrations are too small to be neglected. More importantly, the coefficient of the coma is linearly changed with the yaw angle of the CGH, as shown in Fig. 6(b). Therefore, we can use the linear expression to characterize the relationship between the coefficient of the coma and the yaw angle of CGH. Finally, depending on the departure of the subaperture, we can calculate the yaw angle of CGH for each off-axis subaperture using the calibration result.

 

Fig. 6 The expansion coefficients of the aberrations induced by yawing the CGH cylinder null (a), and the relationship between the coma and the yaw angle of the CGH cylinder null (b).

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4. Stitching approach for acylindrical surface

To minimize the fringe density in the experiment, we set all subapertures at the null configuration, which means that we need to align the best-fitting cylinder axis of the subaperture to the focal line of the diffracted cylindrical wavefront. To accomplish this, the subaperture surface not only needs to be rotated around its best-fitting cylinder axis, but it also requires to be translated in the two-dimensional (2D) plane before measurement. Theoretically, each off-axis subaperture surface can be exactly placed at the null position by following the nominal movement parameters. However, the prerequisite is that we should precisely align and null the center subaperture in advance because the nominal motion parameters of off-axis subapertures are calculated relative to the center subaperture. Unfortunately, owing to the motion error of the multi-axis stage, it is impossible to ensure the actual motion parameters are equal to the nominal motion parameters. For off-axis subapertures, a small gap in the center subaperture will cause their actual motion parameters to significantly deviate from their nominal values. As a result, we cannot accurately determine the corresponding points based on the nominal motion parameters of subapertures.

To determine the corresponding points, we proposed a coarse registration method, which is summarized as follows. First, the subaperture data was transformed into Cartesian coordinates [17]. Here the local coordinate frame is set at the center of the CGH cylinder null. The sub-aperture data measured by interferometer is denoted by phase triplets (u, v, φ, where φ represents the phase value at pixel (u, v.).

Second, according to the nominal motion parameters of subaperture, the local coordinates (x, y, z) yielded from Eq. (4) were transformed into the global coordinate (X, Y, Z) for coarse registration.

Third, based on the data among overlapped areas, the relative motion errors between adjacent subapertures can be calculated with the iterative cylinder-stitching algorithm [19]. Using the calculated motion parameters, all subapertures were coarsely registered with the rigid transformation. In this way, a precise alignment motion is no longer required for each subaperture, and the corresponding points between adjacent subapertures can be determined with the proposed method.

After subtracting the coarse stitching result from the least square fitting radius of every subaperture, we can obtain subaperture data containing small misalignment aberrations and retrace errors. Generally, if the motion parameters of each subaperture are known, the retrace errors induced in each subaperture can be calculated from the simulated system [20–23]. Thanks to the coarse registration process, the actual motion parameters for each subaperture can be calculated from the coarse registration result. Then the retrace errors induced in each subaperture can be calculated from the simulated model drawn in Fig. 2. After subtracting the retrace errors, the subaperture data W contains the figure error and the misalignment aberrations. Different from the cylindrical interferometry [24], the misalignment aberrations induced by tilting and laterally shifting the acylindrical lens are not equal. Therefore, the subaperture data W should be represented as follow,

Consider any two adjacent subapertures indexed as i and j, according to Eq.(7), the phase data in the overlapped areas can be expressed as,

5. Experimental results

We carried out an independent experiment to demonstrate the performance of the proposed method. Figure 7 shows the experimental setup, which consists of a 4″ Zygo GPI/XP interferometer, CGH cylinder null (Diffraction International, H80F3C) with a clear aperture of 80mm × 80mm, and two multi-axes stages. When the CGH cylinder null is set at its normal position, it can diffract a plane wavefront emitted from the interferometer into an f/3 cylindrical wavefront. Interestingly, when it is yawed with a small angle, a varying mount of acylindrical wavefronts can be diffracted. Note that to ensure that the CGH cylinder null is yawed around its centerline, a 2D translation stage is also required to center the CGH cylinder null before the measurement. For details on aligning the center axis of the CGH with the yaw axis, reader can refer to Appendix B. Besides, a multi-axis stage is required to orient the designated surface of the acylindrical lens in front of the interferometric system.

 

Fig. 7 Photograph of the experimental system (a), and the subaperture layout (b).

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The target to be measured is a parabolic cylindrical lens (Thorlabs AYL2018-A), as shown in Fig. 8(a). Its clear aperture is 20mm × 20mm, the conic constant is k = −1, and the ROC at the vertex is ROC = 13.984mm. Based on these parameters, we can calculate its departure from the best-fitting cylinder [21], where the maximum departure is about 81µm, as shown in Fig. 8(b). The large departure makes it difficult to be measured using this CGH cylinder null because the fringes of the off-axis subaperture are too dense, especially in the outermost off-axis subaperture, as shown in Figs. 9(a–d). To obtain a resolvable interferogram at the off-axis subaperture, we yawed the CGH cylinder null with a small angle. It should be noted that to compensate for the coma aberration in the off-axis subaperture, the directions of rotation of the CGH cylinder null and acylindrical lens are opposite to each other.

 

Fig. 8 Picture of the tested acylindrical lens (a) and its departure (b).

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According to the parameters of the CGH and acylindrical lens, 10 off-axis subapertures are required to cover the whole surface of the acylindrical lens, as shown in Fig. 7(b). Owing to the axis symmetry, these off-axis subapertures are divided into five groups. Table 1 lists the nominal motion parameters for each off-axis subaperture. The second row R_c represents the radius of the best-fitting cylinder for each subaperture; the third and fourth rows (y_c, z_c) represent the coordinates of the best-fitting cylinder axis with respect to the vertex of the cylindrical lens. The fifth row α represents the rotation angles of the off-axis subapertures. Using the proposed method, we can also calculate the yaw angle θ of CGH for each off-axis subaperture, as shown in the last row of Table 1. Once the nominal motion parameters have been calculated, we can orient the designated surface in front of the interferometric system and obtain a set of resolvable interferograms, as shown in Figs. 9(e–h). Reader can also refer to Visualization 1 for detail information. Compared with the simulated results given in Figs. 5, the fringe distribution of these measured interferograms agree well with those simulated; deviations may be caused by the misalignment errors and the form error on the tested acylindrical lens (the simulations assume a perfect surface).

Table 1. Nominal motion parameters of off-axis subapertures.

View Table

 

Fig. 9 Interferograms measured at the off-axis subapertures before (a–d) and after (e–h) yawing the CGH cylinder null.

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After collecting data for every subaperture, we firstly performed a coarsely stitching process so as to determine the corresponding points among overlapped regions. Then we used the proposed stitching method to yield a full aperture map of the acylindrical surface, as shown in Fig. 10(a), where the peak-to-valley (PV) and root mean square (RMS) values are 0.891µm and 0.368µm, respectively. To check the repeatability of the proposed method, another measurement was carried out with different misalignment errors of the test acylindrical lens. The result is shown in Fig. 10(b), where the PV and RMS values are 0.902µm and 0.376µm, respectively.

 

Fig. 10 Two stitching results obtained by yawing the cylinder null and the test acylindrical lens was set with different misalignment errors. (a) The first stitching result, (b) the second stitching result.

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To further verify the accuracy of the proposed method, a verification test was carried out, in which the tested acylindrical lens was measured by a contact 3D profilometer (Talysurf PGI Freeform with form accuracy of ±0.15µm, lateral resolution of 0.8nm ) at Taylor Hobson Ltd. AMETEK Ultra Precision Technologies, China. To acquire the figure error with this method, twenty section profiles were first sampling along the circumference direction at 1mm interval. Then, based on these cross-sections, the form deviation was evaluated as 1.05 µm. Compared with the results given in Fig. 10(a), the difference between the results obtained by the two different methods is 0.159µm. In addition, to visually compare the two results, the 3D plot obtained by the Talysurf PGI Freeform is given in Fig. 11(a), and the stitched result given in Fig. 10(a) is converted into a 3D plot, as shown in Fig. 11(b). From these figures, it seems that the results obtained with these two measuring principles have basic conformance. But the newly proposed technique has advantages over the tactile one in being noncontact. Most importantly, we can obtain a high axial resolution form map for the tested acylindrical lens using the proposed method, whereas the readouts from the 3D profilometer are several sectional traces.

 

Fig. 11 Comparison between the 3D profilometer and the proposed method: (a) 3D view of the result measured by the Talysurf PGI Freeform, (b) 3D view of the stitching result obtained by proposed method.

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6. Conclusion

In summary, we proposed a simple and cost-effective stitching interferometry method to acquire the full aperture of a steep acylindrical surface. Experimental results show that the fringe densities of the off-axis subapertures are significantly reduced by yawing the CGH cylinder null. Through a comparison, it is also demonstrated that the proposed technique yields a consistent result with that measured by the Talysurf PGI Freeform, but our scheme is a noncontact method that allows acquisition of the full aperture map with high axial resolution. Different from existing subaperture stitching methods combined with variable null optics, the proposed method does not require an additional auxiliary optics, making the measurement system simple and cost-effective. Furthermore, when combined with the advantages of stitching interferometry, the proposed method may enable us to obtain a full aperture map of large-scale high-numerical-aperture acylindrical optics.

Appendix

A. Mathematical deduction of the Residual aberration of off-axis subapertures

From Eq. (2) given in Section 2, we can solve z as a function of (y),

(10)$$z=a\cdot y+{\left(b_2\cdot y^2+b_1\cdot y+b_0\right)}^{1/2}+c,$$

where

(11)$$\begin{array}{lll}a\hfill & =\hfill & -(k\cdot cos\alpha \cdot sin\alpha )/(1+k\cdot cos{\alpha }^2),\hfill \\ b_2\hfill & =\hfill & [(k+1)(si{n}^4\alpha +co{s}^4\alpha )+2\cdot co{s}^2\alpha \cdot si{n}^2\alpha ]/(1+k\cdot co{s}^2\alpha ),\hfill \\ b_1\hfill & =\hfill & \begin{array}{l}2[(R-z_0\cdot (k+1))\cdot si{n}^3\alpha -y_0\cdot (k+1)co{s}^3\alpha +(R-z_0\cdot (k+1))\cdot co{s}^2\alpha \cdot sin\alpha +\hfill \\ y_0\cdot (k+1)\cdot cos\alpha \cdot si{n}^2\alpha ]/(1+k\cdot co{s}^2\alpha ),\hfill \end{array}\hfill \\ b_0\hfill & =\hfill & \begin{array}{l}[(R-y_0\cdot (k+1))\cdot co{s}^2\alpha +(2R-(k+1)\cdot z_0)\cdot si{n}^2\alpha \hfill \\ +2y_0(R-(k+1)\cdot z_0)\cdot cos\alpha \cdot sin\alpha ]/(1+k\cdot co{s}^2\alpha ),\hfill \end{array}\hfill \\ c\hfill & =\hfill & [R\cdot cos\alpha -z_0\cdot cos\alpha +y_0\cdot sin\alpha -k\cdot z_0\cdot cos\alpha ]/(1+k\cdot co{s}^2\alpha ).\hfill \end{array}$$

Because |α| < 90°, |sin α| < 1, |cos α| < 1. Then cosⁿ⁺¹ α < cosⁿ α, sinⁿ⁺¹ α < sinⁿ α. It is easy to find b₂ ≪ b₁ < b 0 in Eq. (11). Using the Taylor expansion, Eq.(10) can be rewritten as follow,

(12)$$\begin{array}{l}z=a\cdot y+b_0\left(1+1/2\left(b_2/b_0\cdot y^2+b_1/b_0\cdot y\right)-1/4{\left(b_2/b_0\cdot y^2+b_1/b_0\cdot y\right)}^2+\cdots \right)+c\\ =b_0+c+(a+b_1/2)y+(b_2/2-b_1^2/(4b_2))y^2-(b_1{b}_2)/(2b_0)y^3-b_2^2/(4b_0)y^4+\cdots .\end{array}$$

From Eq. (12), it can be seen the coefficients decline quickly as their order become high. The ratio of coefficients between the 4th order and 3th order terms is b₂/(2b₁). Since b₂ ≪ b₁, b₂/(2b₁) is close to zero. As such, we can reasonably truncate Eq. (12) into 3-order polynomials. Then if we subtract the z component of its best fitting cylinder (denoted as z_s), the first and second order terms of Eq. (12) are almost offset. As a result, the departure of the off-axis subaperture is dominated by 3th terms and can be described as follow,

(13)$$z-z_s=m_0+m_{1}y+m_2{y}^2+m_3{y}^3$$

where m₀, m₁ m₂ and m₃ are the coefficients of the residual aberrations.

B. Alignment procedure for the CGH cylinder null

To eliminate the misalignment errors between the center axis of the CGH cylinder null and the yaw axis, we performed an alignment procedure before stitching measurement. A cylindrical surface was taken as the target. The implementation steps are described as follow:

1. Place the cylindrical surface at the null position.
2. Yaw the CGH cylinder null with different angles and obtain the phase data in sequence.
3. To eliminate the effect caused by the figure error of the cylindrical surface, we subtract the result measured at the null position from each measurement.
4. Fit the results obtained from step iii) with a set of 2D Legendre polynomials, then plot the curve (denoted as CA) related with the coefficients of the coma aberration and the yaw angles.

Comparing the CA curves obtained from experiment and simulation, we can determine the misalignment errors of the CGH cylinder null during the yawing process. The reasons are given as below. (1) Since the CGH cylinder null will be set at normal position before rotation, we only need to consider the lateral deviation Δy and the longitudinal deviation Δz, as shown in Fig. 12. (2) For the lateral deviation, it will cause the center of the CGH move in the same direction regardless of forward and reversely yawing the CGH, as shown in Fig. 12(a). Therefore, the sign of the coma aberration induced by the lateral deviation will be unchanged during the yawing process. But the sign of the coma aberration induced by yawing the CGH cylinder null varies with the yaw direction. As such, the absolute values of the coma coefficients will be different if the CGH is yawed with 1° and −1°, respectively. Adopt this feature, we can determine the lateral deviation Δy based on the difference of coma coefficients. (3) Different from the lateral deviation, the longitudinal deviation of the CGH will cause the center of the CGH rotate with the tunable stage, as shown in Fig. 12(b). In that case, the sign of the coma aberration induced by the longitudinal deviation varies with the yaw direction. As a result, the slop of the CA curve obtained from experiment is different from that obtained from the simulation model given in Fig. 2. To determine the longitudinal deviation Δz, we can calculate the difference of the coma coefficients (denoted as ΔDc) obtained from experiment and simulation. On the other hand, it is easy to deduce the relationship between the yaw angle and the coma aberration induced by the longitudinal deviation from Fig. 12(b). Then using this relationship, we can calculate Δz based on ΔDc. (4) With (Δy, Δz), we can eliminate the misalignment errors between the yaw axis and the center axis of the CGH with a precision 2D translation stage.

 

Fig. 12 Misalignment errors of the CGH cylinder null during the yawing process. (a) The center axis of the CGH deviates from the yawing axis with a lateral deviation, (b) the center axis of the CGH deviates from the yawing axis with a longitudinal deviation.

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Funding

Natural Science Foundation of China (NSFC) (61605126); National key research and development program (No.2016YFF0101905); Scientific and Technological Project of the Shenzhen government (JCYJ20150625100821634); Fundamental Research Funds for the Central Universities (21617403).

Acknowledgments

We acknowledge Mr. Tong Zhang from Taylor Hobson Ltd. AMETEK Ultra Precision Technologies, China, for his kind help in measuring the acylindrical lens with a Talysurf PGI FreeForm.

References and links

1. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Optics 49(36), 6924–6929 (2010). [CrossRef]  

2. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small-aperture interferograms,” Opt. Eng. 33(2), 608–613 (1994). [CrossRef]  

3. S. Chen, Y. Dai, S. Li, X. Peng, and J. Wang, “Error reductions for stitching test of large optical flats,” Opt. Laser Technol. 44, 1543–1550 (2012). [CrossRef]  

4. P. Su, J. H. Burge, and R. E. Parks, “Application of maximum likelihood reconstruction of subaperture data for measurement of large flat mirrors,” Appl. Optics 49(1), 21–31 (2010). [CrossRef]  

5. S. Chen, S. Xue, Y. Dai, and S. Li, “Subaperture stitching test of large steep convex spheres,” Opt. Express 23(22), 29047–29058 (2015). [CrossRef]   [PubMed]  

6. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching Interferometry: A Flexible Solution for Surface Metrology,” Optics and Photonics News 14, 38–43 (2003). [CrossRef]  

7. P. Murphy, J. Fleig, F. Forbes, D. Miladinovic, G. DeVries, and S. O’Donohue, “Subaperture stitching interferometry for testing mild aspheres,” Proc. SPIE 6293, 62930J (2006). [CrossRef]  

8. Z. Zhao, H. Zhao, F. Gu, H. Du, and K. Li, “Non-null testing for aspheric surfaces using elliptical sub-aperture stitching technique,” Opt. Express 22(5), 5512–5521 (2014). [CrossRef]   [PubMed]  

9. L. Zhang, D. Liu, T. Shi, Y. Yang, S. Chong, B. Ge, Y. Shen, and J. Bai, “Aspheric subaperture stitching based on system modeling,” Opt. Express 23(15), 19176–19188 (2015). [CrossRef]   [PubMed]  

10. S. Chen, S. Li, Y. Dai, and Z. Zheng, “Lattice design for subaperture stitching test of a concave paraboloid surface,” Appl. Optics 45(10), 2280–2286 (2006). [CrossRef]  

11. M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Annals-Manuf. Techn. 59, 547–550 (2010). [CrossRef]  

12. S. Chen, C. Zhao, Y. Dai, and S. Li, “Reconfigurable optical null based on counter-rotating Zernike plates for test of aspheres,” Opt. Express 22(2), 1381–1386 (2014). [CrossRef]   [PubMed]  

13. D. Liu, T. Shi, L. Zhang, Y. Yang, S. Chong, and Y. Shen, “Reverse optimization reconstruction of aspheric figure error in a non-null interferometer,” Appl. Optics 53(24), 5538–5546 (2014) [CrossRef]  

14. L. Zhang, C. Tian, D. Liu, T. Shi, Y. Yang, H. Wu, and Y. Shen, “Non-null annular subaperture stitching interferometry for steep aspheric measurement,” Appl. Optics 53(25), 5755–5762 (2014) [CrossRef]  

15. J. Peng, X. Liu, X. Peng, Y. Yu, and X. Li, “Selection of F/number in lattice design for stitching inferferometry of aspheric surface,” Proc. SPIE 10250, 1025010 (2016);

16. J. Peng, Q. Wang, X. Peng, and Y. Yu, “Stitching interferometry of high numerical aperture cylindrical optics without using a fringe-nulling routine,” J. Opt. Soc. Am. A 32(11), 1964–1972 (2015). [CrossRef]  

17. S. Chen, S. Li, Y. Dai, and Z. Zheng, “Iterative algorithm for subaperture stitching test with spherical interferometers,” J. Opt. Soc. Am. A , 23(5), 1219–1226 (2006). [CrossRef]  

18. Encoding and fabrication report: CGH cylinder null H80F3C, Tech. Rep. C1437, (Diffraction International, 2014).

19. H. Guo and M. Chen, “Multiview connection technique for 360-deg three-dimensional measurement,” Opt. Eng. 42(4), 900–905 (2003). [CrossRef]  

20. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009) [CrossRef]   [PubMed]  

21. X. Wang, “Measurement of mild asphere by digital plane,” Chin. Opt. Lett. 12(s2), S21201 (2014). [CrossRef]  

22. Q. Hao, S. Wang, Y. Hu, H. Cheng, M. Chen, and T. Li, “Virtual interferometer calibration method of a non-null interferometer for freeform surface measurements,” Appl. Optics 55(35), 9992–10001 (2016). [CrossRef]  

23. T. Shi, D. Liu, Y. Zhou, T. Yan, Y. Yang, L. Zhang, J. Bai, Y. Shen, L. Miao, and W. Huang, “Practical retrace error correction in non-null aspheric testing: A comparison,” Opt. Commun. 383(15), 378–385 (2017). [CrossRef]  

24. J. Peng, H. Xu, Y. Yu, and M. Chen, “Stitching interferometry for cylindrical optics with large angular aperture,” Meas. Sci. Technol. 26(1), 025204 (2015). [CrossRef]