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# Background: Wavefront Interferometry Limitations

Aspheres are widely used in modem optical systems because of their advantage of promising higher image quality with fewer elements compared with traditional spherical systems. Surface figure metrology becomes more challenging as increasingly higher performance demands aspheres with a larger aperture, higher accuracy, and even more complex surface forms, e.g., larger slope variation or a freeform surface.

Wavefront interferometry is a standard solution for the measurement of optical surface error or wavefront aberration, which is usually at the submicron or even nanometer scale. An interferometer fundamentally outputs a test beam and records the fringes formed by interference of the reference beam and the test beam. The reference beam is reflected by a well-polished reference surface, and the test beam is modulated by the test surface or system. The surface error or wavefront aberration is then obtained by analyzing the fringe pattern. Readers are referred to Refs. 1-3 for details of the basic principle of wavefront interferometry if required.

The standard form of the reference surface, which is the last surface of the transmission flat (TF) or transmission sphere (TS) mounted at the exit pupil of the interferometer, is either flat or spherical. Hence, flat or spherical surfaces can be measured in the so-called null test configuration. The name “null test” comes from the fact that a nominally null fringe pattern is obtained because the test wavefront perfectly matches the test surface. However, for aspheres, non-null fringes are, of course, observed, and the number of fringes depends on the aspheric departure from the best-fit sphere of the surface. On the other hand, the number of resolvable fringes of the interferometer is limited by the Nyquist frequency. The dynamic range of measurement of a commercial interferometer is typically restricted to only tens of fringes.

Worse still, the lateral range of measurement is also strictly limited by the aperture of the interferometer when testing flat or convex surfaces as the test beam is collimated or convergent. For example, convex spheres are null tested at confocal positions, i.e., the center of curvature of the test surface coincides with the focus of the TS, as shown in Fig. 1. The lateral range of measurement is obviously limited by the f/number (the ratio of the focal length to the diameter) of the TS, which should be smaller than the R/number (the ratio of the radius of curvature to the diameter) of the test surface. The smallestf/number of the TS commercially available isf/0.65. As a matter of fact, we cannot test a convex hemisphere or full sphere directly, no matter how small the diameter is.

This fact is really disillusioning because wavefront interferometry is restricted to only a small group in the family of modern optics. Great efforts have been made to extend both the lateral and dynamic ranges of measurement, including null and non-null tests with auxiliary optics. Subaperture stitching interferometry solves this problem by dividing the full aperture into a series of smaller

Figure 1 Null test of spherical surface at the confocal position.

subapertures that overlap with their neighbors, as shown in Fig. 2. The subaper- tues are measured one by one and finally stitched together to give a full-aperture surface error map. This method has three advantages. First, very large apertures can be measured now because what we need to do is to measure small pieces of subapertures. Second, aspheres with larger aspheric departure can be measured because a subaperture’s departure is reduced to generate fringes resolvable with a standard interferometer. Finally, as a small subaperture is spotlighted, more details of the surface error can be resolved by the interferometer. Therefore, a surface error of higher spatial frequency can be obtained by subaperture stitching interferometry.

Figure 2 Schematic diagram of subaperture stitching interferometry.

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