Johannes A. Soons

A simple ball averager for reference sphere calibrations

When measuring the form errors of precision optics with an interferometer, calibration of the reference wavefront is of central importance. In recent years, ball averaging, or random ball testing, has emerged as a robust method for calibrating spherical reference wavefronts in converging beams. We describe a simple instrument, consisting of an air bearing and two electric motors, that can rotate the test ball around three axes as required for a ball averaging test. The performance of the instrument is demonstrated by using it to calibrate a concave transmission sphere. Further we discuss the effects of image sampling at random locations or on uniform grids, and the effect of correlated measurements. Finally, we describe a method to determine the number of measurements which are sufficient for a ball averaging calibration.

Three-flat tests including mounting-induced deformations

We investigate three-flat calibration methods based on rotational symmetry and mirror symmetry for absolute interferometric flatness measurements of circular flats in the presence of deformations caused by the support mechanism for the flats, which are a significant problem for large, heavy flats. We show that the mirror-symmetric component of the mounting-induced deformation can be determined by comparing flat test solutions based on mirror symmetry and on rotational symmetry, when the flats have identical deformations. We also describe a novel solution to the three-flat problem for three flats with identical mounting-induced deformations. In the new three-flat solution, the flat deformation is calculated along with the wavefront flatness errors for the three flats. Formulas for the uncertainty of three-flat test solutions are derived.

Measuring Form and Radius of Spheres with Interferometry

Abstract The geometry of a nearly spherical surface, for example that of a precision optic, is completely determined by the radius-of-curvature at one point and the deviation from the perfect spherical form at all other points of the sphere. Measurements of radius and form error can now be made with interferometers to remarkable accuracy. We describe measurements of radius and form error of a precision silicon sphere, having a nominal radius of 46.8 mm, with the “extremely accurate CALIBration InterferometeR” (XCALIBIR) at the National Institute of Standards and Technology (NIST). For these measurements XCALIBIR is configured as a spherical Fizeau interferometer providing a field of view of 44°. To measure the radius, a variant of the well known interferometric radius bench method is used. Careful alignment of phase measuring and displacement measuring interferometers enables us to achieve a standard measurement uncertainty for the sphere radius of about 5 parts in 10 7 . The measurement of the form error is complicated because the entire sphere surface cannot be imaged in one measurement. Instead, 138 overlapping areas of the sphere surface are measured. A “stitching” algorithm is then employed to assemble these measurements into a form error map for the entire surface. We show that form errors can lead to considerable uncertainty in the radius of a sphere obtained through a radius-of-curvature measurement with the radius bench method.

On the Geometric and Thermal Errors of a Hexapod Machine Tool

This paper describes the measurement and analysis of the quasi-static errors of a prototype hexapod milling machine located at the National Institute of Standards and Technology (NIST). Emphasis is placed on a) identification of the most important parametric errors and their potential impact on performance, b) description of the techniques used to measure and estimate the parametric errors, and c) application of various concepts to model the geometric and thermal errors of the hexapod.

Absolute interferometric tests of spherical surfaces based on rotational and translational shears

Traceability of interferometric form measurements requires characterization of the reference wavefront. We investigate absolute tests for spherical surfaces where the form errors of both reference and test surface are estimated by minimizing the difference in measurements obtained at various positions and orientations of the test surface. This approach yields an estimate for the test surface errors without changing experimental settings, such as cavity length, that may affect the apparent reference errors. The method requires at least one translation of the test surface in the field of view and one rotation. Additional measurements provide redundancies to improve and characterize measurement uncertainties. The errors of the reference and test surface are estimated with pixel-level spatial resolution without assuming an underlying error model, such as a representation based on Zernike polynomials. The estimation algorithm consists of an iterative sequence of stitching steps, with the role of reference and test surface reversed for each step. Measurement uncertainties are evaluated using Monte-Carlo procedures and analysis of residual errors for experiments with redundant measurements. Key sources of measurement uncertainty are spatially correlated measurement errors resulting from errors in test surface location, image distortion, and environmental effects. Experimental results are presented comparing the method to a random ball test.

Development of a metrology frame to improve the positioning accuracy of micro/meso-scale machine tools

The small work volumes of Micro/Meso-scale Machine Tools (MMMTs) often present problems for calibration and error compensation, but also allow solutions not practical on the traditional scale. Measuring tool position with a separate metrology frame and compensating for error motions is one such solution. The metrology frame design follows principles of precision design and allows measurement of the position of the tool tip with respect to the workpiece while minimising Abbe errors. Kinematic analysis provides the relationship between metrology frame measurements and machine tool coordinates. Error analysis reveals that sensor error has the only first order influence on measurement accuracy.

Uncertainties in aspheric profile measurements with the geometry measuring machine at NIST

The Geometry Measuring Machine (GEMM) of the National Institute of Standards and Technology (NIST) is a profilometer for free-form surfaces. A profile is reconstructed from local curvature of a test part surface, measured at several locations along a line. For profile measurements of free-form surfaces, methods based on local part curvature sensing have strong appeal. Unlike full-aperture interferometry they do not require customized null optics. The uncertainty of a reconstructed profile is critically dependent upon the uncertainty of the curvature measurement and on curvature sensor positioning. For an instrument of the GEMM type, we evaluate the measurement uncertainties for a curvature sensor based on a small aperture interferometer and then estimate the uncertainty in the reconstructed profile that can be achieved. In addition, profile measurements of a free-form mirror, made with GEMM, are compared with measurements using a long-trace profiler, a coordinate measuring machine, and subaperture-stitching interferometry.

Deformation-free form error measurement of thin, plane-parallel optics floated on a heavy liquid

We describe a novel method for measuring the unconstrained flatness error of thin, plane-parallel precision optics. Test parts are floated on high-density aqueous metatungstate solutions while measuring the flatness error with an interferometer. The support of the flat optics by the uniform hydrostatic pressure at the submerged face of the flat optic eliminates flatness errors caused by mounting forces. A small, well characterized flatness error results from the bending of the floating flat by the hydrostatic pressure gradient at the edges. An equation describing the bending of thin, flat plates floating on a liquid is derived, which can be used to correct the flatness measurements of arbitrarily shaped plates. The method can be used to measure flatness errors of both nontransparent and transparent parts, and it is illustrated with flatness measurements of photomask blanks and substrates for extreme ultraviolet lithography. The refractive index of a saturated aqueous lithium metatungstate solution was measured at 632.8 nm and was found to be close to the refractive indices of several low thermal expansion optical materials.

Form-Profiling of Optics Using the Geometry Measuring Machine and the M-48 CMM at NIST

We are developing an instrument, the Geometry Measuring Machine (GEMM), to measure the profile errors of aspheric and free form optical surfaces, with measurement uncertainties near 1 nm. Using GEMM, an optical profile is reconstructed from local curvatures of a surface, which are measured at points on the optic’s surface. We will describe a prototype version of GEMM, its repeatability with time, a measurements registry practice, and the calibration practice needed to make nanometer resolution comparisons with other instruments. Over three months, the repeatability of GEMM is 3 nm rms, and is based on the constancy of the measured profile of an elliptical mirror with a radius of curvature of about 83 m. As a demonstration of GEMM’s capabilities for curvature measurement, profiles of that same mirror were measured with GEMM and the NIST Moore M-48 coordinate measuring machine. Although the methods are far different, two reconstructed profiles differ by 22 nm peak-to-valley, or 6 nm rms. This comparability clearly demonstrates that with appropriate calibration, our prototype of the GEMM can measure complex-shaped optics.

Holographic radius test plates for spherical surfaces with large radius of curvature

We describe a novel interferometric method, based on nested Fresnel zone lenses or photon sieves, for testing and measuring the radius of curvature of precision spherical surfaces that have radii in a range between several meters and a few hundred meters. We illustrate the measurement concept with radius measurements of a spherical mirror with a radius of about 10 m. The measured radius is 9877  mm±10  mm for a coverage factor k=2. Our measurements also demonstrate, for the first time to the best of our knowledge, the utility of photon sieves for precision surface metrology because they diffuse higher diffraction orders of computer generated holograms, which reduces coherent noise.