# Large convex spheres

Large convex and/or small R/number spheres have more applications in modern optical systems, such as the lithographic lens^{73} and large telescopes. In telescope design, spherical and aspheric lenses are usually used as correctors near the focal plane. For example, the Large Synoptic Survey Telescope includes a refractive camera design with three lenses and a set of filters.^{74} One of the lenses has a convex sphere of 1.55 m in diameter and 2.824 m in radius of curvature.

This case focuses on the interferometric test of the half meter-class convex sphere shown in Fig. 29. The clear aperture is 506 mm, the R/number is smaller than R/0.61, and the angle between the optical axis and the normal at the edge approaches 55 deg. It is a near hemisphere. To avoid coupling with the material inhomogeneity, we have to reject the transmitted wavefront test method. A subaperture stitching test is employed instead to break the limits of both aperture and f/number by dividing the full aperture into a series of smaller subapertures.

**Figure 29 **Drawing of the steep convex sphere under test.

Serious problems still arise in the stitching test of such a large, steep surface. Because the subaperture is much smaller than the full aperture, a self-calibrated stitching algorithm is required to separate the TS reference error. Because of the strong nonlinearity of the coordinate mapping, we need to further process the stitching result to get the true surface error rather than directly removing power, as we usually do.

The subaperture test is still in a confocal null test configuration, so the TS radius must be longer than that of the test surface (308 mm). We select the 6-in. f/3.5 TS with a radius of 475.8 mm. The size of the subaperture is about 88 mm, and we need a total of 90 off-axis subapertures arranged on five rings around the central one, as shown in Fig. 30. The off-axis angles are 10, 19, 29, 39, and 49 deg, respectively. Subapertures are evenly spaced on each ring. Because of the big off-axis angle at the edge, the footprint of the outermost subaperture on the *OXY* plane is elliptical.

**Figure 30 **Subaperture layout.

**Figure 31 **Experimental setup.

The experimental setup is shown in Fig. 31. During the subaperture test, the test surface is moved to the subaperture’s nominal position by yawing and rotating, while the three translational axes are responsible for fine aligning and nulling the interferometer.

We apply the self-calibrated stitching algorithm and get the calibrated reference error described in 37-term Zernike polynomials. The reference error map shown in Fig. 32(a) is approximately consistent with Zygo’s final report of the TS, shown in Fig. 32(b).

Readers may worry about the coupling of the systematic error with the surface error. However, in this algorithm, the surface error is unaffected by introducing additional Zernike terms during stitching, which is indirectly verified by the consistency of self-calibrated TS reference error and Zygo’s report. On one hand, only those components of the surface error giving identical subaperture errors are not differentiable from systematic errors. For example, the spherical aberration of rotational symmetry gives identical subaperture errors for those subapertures lying on the same ring (see Fig. 30), but there is no such component giving identical errors for all subapertures with different off-axis distances. Moreover, the identical subaperture error is composed of certain terms with certain proportions. For example, spherical aberration in the full aperture typically contains coma and astigmatism with certain proportions when masked in a local off-axis subaperture. Therefore, the surface error will not be smoothed out by fitting extra Zernike polynomials because we use the same systematic error form for all subapertures. On the other hand, the stitching process does not add extra Zernike terms to the surface error either. Because the QR decomposition used to solve the LS problem *Ax = b* gives a solution with at most *k* nonzero elements, where *k* is the column rank of matrix A. If A has full column rank, then the solution has the minimal norm. This mathematical background indicates that no extra Zernike terms will be added as a systematic error to each subaperture, even if they do not contribute to the overlapping mismatch error.

**Figure 32 **Calibrated TS reference error: (a) self-calibrated result and (b) Zygo’s final report (PVr and Zernike fit).

Usually, when we get the error map for spherical or aspheric surfaces, we remove the piston, tip-tilt, and power to evaluate the true surface error regardless of the uncertainty of the radius of curvature. However, a serious problem arises when we do that for this very small R/number sphere. It stems from the strong nonlinearity of coordinate mapping. The lateral coordinates in a subaperture’s local frame is related to those in the full aperture’s global frame as follows:

The azimuthal angle у = 0 for off-axis subapertures on the x axis. Equation (44) indicates the compression in the x coordinate. As a result, the power in the global frame produces astigmatism in the local frame:

For test surfaces with a big R/number, the off-axis angle is relatively small, and the astigmatism in Eq. (45) can be neglected. For example, the coefficient of astigmatism is only 0.015 for p = 10 deg. However, for the test surface here, the angle of the outmost subaperture is p = 49 deg, and the coefficient of astigmatism is about 0.285. The strong nonlinearity generates incorrect subaperture astigmatism and finally leads to significantly incorrect spherical aberration in the full aperture map. Post-processing of the stitching result is necessary to get the true surface error rather than directly removing power, as we usually do. We propose to assess the sphericity by removing the kinematic error in rigid body transformation. It is now well developed in the field of computer-aided tolerancing. Mathematically, the optimal configuration g e Q must be found such that the following function is minimized:

where d(g,p) is the signed distance of point p to the best-fit sphere, and C is a slack variable dealing with the radius uncertainty. Note that the transformation should be

**Figure 33 **Stitching results with sphericity assessment (a) before figuring and (b) after figuring.

**Figure 34 **Near-null subaperture test with counter-rotating CGHs: (a) CGH in mount and (b) setup of near-null subaperture test.

**Figure 35 **Near-null subaperture wavefronts: (a) center (PV 2.522X, RMS 0.338X), (b) ring 1 (PV 3.035X, RMS 0.603X), (c) ring 2 (PV 5.359X, RMS 0.905X), and (d) ring 3 (PV 4.015X, RMS 0.707X).

acted on Cartesian coordinates of measuring point p, which are calculated by adding the signed deviation (given by the stitched result) in the normal direction to the corresponding points on the nominal sphere. The configuration space Q for sphericity assessment is simply a 3-D translational group, i.e., we need only search for the optimal transformation in three orthogonal translations. Details of the algorithm for general geometric tolerances can be found in Ref. 75.

The stitched map of sphericity after post-processing is shown in Fig. 33(a). We use this map for corrective figuring and finally obtain the error map shown in Fig. 33(b). The PV and RMS values are greatly reduced, which also verifies the stitching method.