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# Hyper-hemispheres

A hyper-hemisphere is referred to as a surface portion on a sphere with a solid angle exceeding 2n. It can find many applications in precision components and instruments such as gyro rotors,70,71 precision ball bearings,72,73 and calibration spheres.20 The figure error or the sphericity of the whole surface is vital for the instruments to achieve a high performance. For example, the rotor in the electrostatic-suspended gyro has a sphericity of ±12.5 nm to meet the extremely high demands of the GP-B project.71 A hyper-hemisphere can generally be inspected with a high-precision roundness measurement.70-72 By using an airbearing spindle and the spindle error separation technique, the roundness- measuring instrument can achieve an ultra-high-precision measurement. However, it is difficult to tie all roundness measurements together within a tight tolerance because roundness measurements at different locations and orientations invite mechanical motion error. Moreover, due to under-sampling, multiple roundness measurements are not sufficient to reveal the sphericity of the whole surface.

Actually, many precision hyper-hemispheres, including full spheres, can be considered optical surfaces, i.e., they have good surface quality to be interfero- metrically tested. Nevertheless, limited by thef/number of available TSs (typically no less than f/0.65), the interferometer can measure only a small portion on the sphere. It is impossible for the test beam to cover the whole surface of a hyperhemisphere. Therefore, a subaperture stitching method was naturally proposed to extend the coverage area.19,20

For hyper-hemispheres, a special method is required to determine the overlapping correspondence. The projection ambiguity arises when all measurement points are projected onto the nominal sphere, and then the projections are again projected onto the equator plane, as is usually done for the overlapping calculation subproblem. In this plane, the envelope of projections determines the overlap region. However, a couple of reciprocal points located on the southern and northern hemispheres, respectively, may be projected to the same point on the equator plane. Such a projection ambiguity arises from the Cartesian coordinates used to describe points on a hyper-hemisphere. It can be avoided by using the coordinates of latitude /a and longitude /o because there is one-to-one mapping between the coordinates and the points on a sphere, except for the two poles. Thus, the projections on the nominal sphere are projected onto the plane of latitude and longitude, rather than the equator plane. The projection area on the plane of latitude and longitude is restricted within [ -п, п] x [ -п/2, п/2], as shown in Fig. 26.

This case shows the subaperture stitching test of a hemisphere and a full sphere with a 38-mm spherical diameter. An f/0.75 TS is chosen with about a 25-mm illuminated subaperture. For the hemisphere, seven subapertures are simply arranged, including six peripheral subapertures plus the central one. The angle

Figure 26 Projection onto the plane of latitude and longitude.

between the optical axis of the peripheral one and that of the central one is 60 deg. For the full sphere, we propose to test it with three annular strips wrapping the sphere, easily implemented by two orthogonal rotations. As shown in Fig. 26, the six subapertures are evenly distributed on the annular strip, and the three strips contain the same two polar subapertures. Therefore, there are a total of 14 subapertures.

Because the subaperture surface error is comparable to the reference surface error, the overlapping inconsistency is shown in Fig. 27(a), as subaperture traces are obvious in the stitched map. By applying the self-calibrated stitching algorithm, the real surface error is obtained without visible overlapping traces.

Figure 27 Stitched error map of the hemisphere (a) with reference error included, (b) with reference error self-calibrated, and (c) after ion beam figuring.

Figure 28 Stitched error map of the full sphere (a) before figuring and (b) after figuring.

As shown in Fig. 27(b), the initial error is 130.1 nm PV and 12.7 nm RMS. This error is then corrected by the ion-beam figuring process. The surface error is reduced to 110.2 nm PV and 11.5 nm RMS, as shown in Fig. 27(c). The self- calibrated reference error is similar to the factory report of the TS from Zygo that indirectly verifies the stitching method.

For the full sphere, we prefer to present the error distribution in a globe manner to avoid the projection ambiguity. The peaks and valleys are attached to a globe that can be dynamically rotated to show the error distribution in different orientations. By applying the self-calibrated stitching algorithm, the initial sphericity is shown in Fig. 28(a), and the sphericity after figuring is shown in Fig. 28(b). The error is reduced from 8.3 nm RMS to 6.2 nm RMS.

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