# Measurement of High Numerical Aperture Cylindrical Surface with Iterative Stitching Algorithm
## 4 Experimental Demonstration  
Experimental measurements were carried out to verify the practicality of the proposed algorithm. The experimental arrangement consisted of a Fizeau interferometer (Zygo GPI/XP 4, Zygo, Middlefield, CT, USA), a CGH cylinder null (H80F3C, Diffraction international, Minnetonka, MN, USA) and a multi-dimensional adjustment stage (Figure 5). A CGH cylinder null, 80 mm high \(\times\) 80 mm wide, was used as the cylindrical wavefront converter of the interference system. The maximum measurable aperture angle was nearly 18.9\({}^{\circ}\) (the treatment of the image borders led to a measured angle less than 20 degrees). The cylindrical lens was 53 mm high \(\times\) 50.8 mm wide and curvature radius was 26.4 mm. For obtaining the whole surface of the cylindrical lens, we collected 13 sub-apertures and set the rotation angle to 12\({}^{\circ}\). We did not strictly align the lens, so after each rotation, the fringe of interferogram was too dense to be analyzed. So, we manually tilted and leveled the sub-aperture to obtain an interferogram that was near to non-fringe. Then, the remaining position errors were calculated through the iterations. Figure 6 shows the phase maps and interferograms of the acquired sub-aperture. In order to meet the condition of the null test, we ensured that the fringe was small enough to make the ideal cylindrical wavefront and cylindrical lens coincide. The remaining fringes were caused by the contour deviation and remaining position errors. Figure 4: (**a**) The stitching result without using the iteration algorithm. (**b**) The result obtained with the proposed iterative method, (**c**) the residual error between the origin map and traditional method, and (**d**) the residual error between the origin map and iterative method. The stitching result map without the iterations is depicted in Figure 7a, where the stitching marks can be seen. Accordingly, the result map after three iterations (Figure 7b) was seamless, the PV value of the stitching result was 1.003 \(\upmu\)m, and the RMS value was 0.174 \(\upmu\)m. To evaluate the effectiveness of the proposed stitching method, we observed the variation in the mismatch map within the overlapping regions, as shown in Figure 7c,d. The variation was determined by calculating the radial deviations of the corresponding points in the overlapping region, and can be thought of as the residual noise following stitching process. The PV value of the residual error without iterations was 0.619 \(\upmu\)m and the RMS value was 0.029 \(\upmu\)m. The RMS value obtained with the proposed method was 0.005 \(\upmu\)m. This confirms that the proposed stitching model can eliminate the discrepancies among the overlapping regions. The invalid points in Figure 7a were caused by the spurious fringe in the central sub-aperture. Due to the interpolation process in the coordinate transformation, there was no invalid point in the middle part of the stitching result shown in Figure 7b. In order to evaluate the reliability of the proposed algorithm, we carried out 10 iterations of the experimental data and compared the coefficients of the sub-aperture adjustment for each iteration of the two sub-apertures. The experimental data (Figure 8) showed that after three iterations, the \(y\)-direction and \(z\)-direction errors were stabilized within a very small range. From the graph, for the whole experimental system and its operation, the rotation errors around the \(Y\) and \(Z\) axes were relatively small. Figure 5: (**a**) Photograph of the experimental system and (**b**) the experimental sample. Figure 6: (**a**) Measured interferograms of different sub-apertures, fringe maps of sub-apertures and (**b**) the phase maps of sub-apertures. Table 2 summaries the position error \(Err_{i}^{k}\) of three iterations and Figure 9 shows an interferogram of \(Err_{i}^{k}\). As the ninth sub-aperture was chosen as the benchmark, the 24\({}^{\circ}\) interferogram is not shown.